arXiv:1612.09375v1
[math.CT]
30
Dec
2016
Basic
Category
Theory
TOM
LEINSTER
University
of
Edinburgh
First
published
as
Basic
Category
Theory,
Cambridge
Studies
in
Advanced
Mathematics,
Vol.
143,
Cambridge
University
Press,
Cambridge,
2014.
ISBN
978-1-107-04424-1
(hardback).
Information
on
this
title:
http://www.cambridge.org/9781107044241
c
Tom
Leinster
2014
This
arXiv
version
is
published
under
a
Creative
Commons
Attribution-NonCommercial-ShareAlike
4.0
International
licence
(CC
BY-NC-SA
4.0).
Licence
information:
https://creativecommons.org/licenses/by-nc-sa/4.0
c
Tom
Leinster
2014,
2016
Preface
to
the
arXiv
version
This
book
was
first
published
by
Cambridge
University
Press
in
2014,
and
is
now
being
published
on
the
arXiv
by
mutual
agreement.
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http://www.cambridge.org/
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editable.
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stance,
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but
some
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remove
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or
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own.
Similarly,
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Attribution-
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(CC
BY-NC-SA
4.0).
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lows
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A
TEX
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the
printed
version
except
for
the
correction
of
a
small
number
of
minor
errors.
Thanks
to
all
those
who
pointed
these
out,
in-
cluding
Martin
Brandenburg,
Miguel
Couto,
Bradley
Hicks,
Thomas
Moeller,
and
Yaokun
Wu.
Tom
Leinster,
December
2016
iii
Contents
page
vii
1
Note
to
the
reader
Introduction
1
Categories,
functors
and
natural
transformations
1.1
Categories
1.2
Functors
1.3
Natural
transformations
9
10
17
27
2
Adjoints
2.1
Definition
and
examples
2.2
Adjunctions
via
units
and
counits
2.3
Adjunctions
via
initial
objects
41
41
50
58
3
Interlude
on
sets
3.1
Constructions
with
sets
3.2
Small
and
large
categories
3.3
Historical
remarks
65
66
73
78
4
Representables
4.1
Definitions
and
examples
4.2
The
Yoneda
lemma
4.3
Consequences
of
the
Yoneda
lemma
83
84
93
99
5
Limits
5.1
Limits:
definition
and
examples
5.2
Colimits:
definition
and
examples
5.3
Interactions
between
functors
and
limits
107
107
126
136
6
Adjoints,
representables
and
limits
6.1
Limits
in
terms
of
representables
and
adjoints
6.2
Limits
and
colimits
of
presheaves
6.3
Interactions
between
adjoint
functors
and
limits
142
142
146
158
Appendix
Proof
of
the
general
adjoint
functor
theorem
171
174
177
178
Further
reading
Index
of
notation
Index
v
Note
to
the
reader
This
is
not
a
sophisticated
text.
In
writing
it,
I
have
assumed
no
more
mathe-
matical
knowledge
than
might
be
acquired
from
an
undergraduate
degree
at
an
ordinary
British
university,
and
I
have
not
assumed
that
you
are
used
to
learn-
ing
mathematics
by
reading
a
book
rather
than
attending
lectures.
Furthermore,
the
list
of
topics
covered
is
deliberately
short,
omitting
all
but
the
most
funda-
mental
parts
of
category
theory.
A
‘further
reading’
section
points
to
suitable
follow-on
texts.
There
are
two
things
that
every
reader
should
know
about
this
book.
One
concerns
the
examples,
and
the
other
is
about
the
exercises.
Each
new
concept
is
illustrated
with
a
generous
supply
of
examples,
but
it
is
not
necessary
to
understand
them
all.
In
courses
I
have
taught
based
on
earlier
versions
of
this
text,
probably
no
student
has
had
the
background
to
understand
every
example.
All
that
matters
is
to
understand
enough
examples
that
you
can
connect
the
new
concepts
with
mathematics
that
you
already
know.
As
for
the
exercises,
I
join
every
other
textbook
author
in
exhorting
you
to
do
them;
but
there
is
a
further
important
point.
In
subjects
such
as
number
theory
and
combinatorics,
some
questions
are
simple
to
state
but
extremely
hard
to
answer.
Basic
category
theory
is
not
like
that.
To
understand
the
question
is
very
nearly
to
know
the
answer.
In
most
of
the
exercises,
there
is
only
one
possible
way
to
proceed.
So,
if
you
are
stuck
on
an
exercise,
a
likely
remedy
is
to
go
back
through
each
term
in
the
question
and
make
sure
that
you
understand
it
in
full.
Take
your
time.
Understanding,
rather
than
problem
solving,
is
the
main
challenge
of
learning
category
theory.
Citations
such
as
Mac
Lane
(1971)
refer
to
the
sources
listed
in
‘Further
reading’.
This
book
developed
out
of
master’s-level
courses
taught
several
times
at
the
University
of
Glasgow
and,
before
that,
at
the
University
of
Cambridge.
In
turn,
the
Cambridge
version
was
based
on
Part
III
courses
taught
for
many
vii
viii
Note
to
the
reader
years
by
Martin
Hyland
and
Peter
Johnstone.
Although
this
text
is
significantly
different
from
any
of
their
courses,
I
am
conscious
that
certain
exercises,
lines
of
development
and
even
turns
of
phrase
have
persisted
through
that
long
evo-
lution.
I
would
like
to
record
my
indebtedness
to
them,
as
well
as
my
thanks
to
François
Petit,
my
past
students,
the
anonymous
reviewers,
and
the
staff
of
Cambridge
University
Press.
Introduction
Category
theory
takes
a
bird’s
eye
view
of
mathematics.
From
high
in
the
sky,
details
become
invisible,
but
we
can
spot
patterns
that
were
impossible
to
de-
tect
from
ground
level.
How
is
the
lowest
common
multiple
of
two
numbers
like
the
direct
sum
of
two
vector
spaces?
What
do
discrete
topological
spaces,
free
groups,
and
fields
of
fractions
have
in
common?
We
will
discover
answers
to
these
and
many
similar
questions,
seeing
patterns
in
mathematics
that
you
may
never
have
seen
before.
The
most
important
concept
in
this
book
is
that
of
universal
property.
The
further
you
go
in
mathematics,
especially
pure
mathematics,
the
more
universal
properties
you
will
meet.
We
will
spend
most
of
our
time
studying
different
manifestations
of
this
concept.
Like
all
branches
of
mathematics,
category
theory
has
its
own
special
vo-
cabulary,
which
we
will
meet
as
we
go
along.
But
since
the
idea
of
universal
property
is
so
important,
I
will
use
this
introduction
to
explain
it
with
no
jargon
at
all,
by
means
of
examples.
Our
first
example
of
a
universal
property
is
very
simple.
Example
0.1
Let
1
denote
a
set
with
one
element.
(It
does
not
matter
what
this
element
is
called.)
Then
1
has
the
following
property:
for
all
sets
X,
there
exists
a
unique
map
from
X
to
1.
(In
this
context,
the
words
‘map’,
‘mapping’
and
‘function’
all
mean
the
same
thing.)
Indeed,
let
X
be
a
set.
There
exists
a
map
X
→
1,
because
we
can
define
f
:
X
→
1
by
taking
f
(x)
to
be
the
single
element
of
1
for
each
x
∈
X.
This
is
the
unique
map
X
→
1,
because
there
is
no
choice
in
the
matter:
any
map
X
→
1
must
send
each
element
of
X
to
the
single
element
of
1.
Phrases
of
the
form
‘there
exists
a
unique
such-and-such
satisfying
some
1
2
Introduction
condition’
are
common
in
category
theory.
The
phrase
means
that
there
is
one
and
only
one
such-and-such
satisfying
the
condition.
To
prove
the
existence
part,
we
have
to
show
that
there
is
at
least
one.
To
prove
the
uniqueness
part,
we
have
to
show
that
there
is
at
most
one;
in
other
words,
any
two
such-and-
suches
satisfying
the
condition
are
equal.
Properties
such
as
this
are
called
‘universal’
because
they
state
how
the
ob-
ject
being
described
(in
this
case,
the
set
1)
relates
to
the
entire
universe
in
which
it
lives
(in
this
case,
the
universe
of
sets).
The
property
begins
with
the
words
‘for
all
sets
X’,
and
therefore
says
something
about
the
relationship
between
1
and
every
set
X:
namely,
that
there
is
a
unique
map
from
X
to
1.
Example
0.2
This
example
involves
rings,
which
in
this
book
are
always
taken
to
have
a
multiplicative
identity,
called
1.
Similarly,
homomorphisms
of
rings
are
understood
to
preserve
multiplicative
identities.
The
ring
Z
has
the
following
property:
for
all
rings
R,
there
exists
a
unique
homomorphism
Z
→
R.
To
prove
existence,
let
R
be
a
ring.
Define
a
function
φ
:
Z
→
R
by
1
|
+
{z
·
·
·
+
}
1
if
n
>
0,
n
φ(n)
=
0
if
n
=
0,
−φ(−n)
if
n
<
0
(n
∈
Z).
A
series
of
elementary
checks
confirms
that
φ
is
a
homomorphism.
To
prove
uniqueness,
let
R
be
a
ring
and
let
ψ
:
Z
→
R
be
a
homomorphism.
We
show
that
ψ
is
equal
to
the
homomorphism
φ
just
defined.
Since
homomor-
phisms
preserve
multiplicative
identities,
ψ(1)
=
1.
Since
homomorphisms
preserve
addition,
ψ(n)
=
ψ(1
·
·
·
+
}
1
)
=
ψ(1)
+
·
·
·
+
ψ(1)
=
|
1
+
{z
·
·
·
+
}
1
=
φ(n)
|
+
{z
|
{z
}
n
n
n
for
all
n
>
0.
Since
homomorphisms
preserve
zero,
ψ(0)
=
0
=
φ(0).
Finally,
since
homomorphisms
preserve
negatives,
ψ(n)
=
−ψ(−n)
=
−φ(−n)
=
φ(n)
whenever
n
<
0.
Crucially,
there
can
be
essentially
only
one
object
satisfying
a
given
univer-
sal
property.
The
word
‘essentially’
means
that
two
objects
satisfying
the
same
universal
property
need
not
literally
be
equal,
but
they
are
always
isomorphic.
For
example:
Lemma
0.3
Let
A
be
a
ring
with
the
following
property:
for
all
rings
R,
there
exists
a
unique
homomorphism
A
→
R.
Then
A
Z.
Introduction
3
Proof
Let
us
call
a
ring
with
this
property
‘initial’.
We
are
given
that
A
is
initial,
and
we
proved
in
Example
0.2
that
Z
is
initial.
Since
A
is
initial,
there
is
a
unique
homomorphism
φ
:
A
→
Z.
Since
Z
is
initial,
there
is
a
unique
homomorphism
φ
0
:
Z
→
A.
Now
φ
0
◦
φ
:
A
→
A
is
a
homomorphism,
but
so
too
is
the
identity
map
1
A
:
A
→
A;
hence,
since
A
is
initial,
φ
0
◦
φ
=
1
A
.
(This
follows
from
the
uniqueness
part
of
initiality,
taking
‘R’
to
be
A.)
Similarly,
φ
◦
φ
0
=
1
Z
.
So
φ
and
φ
0
are
mutually
inverse,
and
therefore
define
an
isomorphism
between
A
and
Z.
This
proof
has
very
little
to
do
with
rings.
It
really
belongs
at
a
higher
level
of
generality.
To
properly
understand
this,
and
to
convey
more
fully
the
idea
of
universal
property,
it
will
help
to
consider
some
more
complex
examples.
Example
0.4
Let
V
be
a
vector
space
with
a
basis
(v
s
)
s∈S
.
(For
example,
if
V
is
finite-dimensional
then
we
might
take
S
=
{1,
.
.
.
,
n}.)
If
W
is
another
vector
space,
we
can
specify
a
linear
map
from
V
to
W
simply
by
saying
where
the
ba-
sis
elements
go.
Thus,
for
any
W,
there
is
a
natural
one-to-one
correspondence
between
linear
maps
V
→
W
and
functions
S
→
W.
This
is
because
any
function
defined
on
the
basis
elements
extends
uniquely
to
a
linear
map
on
V.
Let
us
rephrase
this
last
statement.
Define
a
function
i
:
S
→
V
by
i(s)
=
v
s
(s
∈
S
).
Then
V
together
with
i
has
the
following
universal
property:
S
∀
functions
f
i
/
V
∃!
linear
f
¯
!
∀W.
This
diagram
means
that
for
all
vector
spaces
W
and
all
functions
f
:
S
→
W,
there
exists
a
unique
linear
map
f
¯
:
V
→
W
such
that
f
¯
◦
i
=
f
.
The
symbol
∀
means
‘for
all’,
and
the
symbols
∃!
mean
‘there
exists
a
unique’.
Another
way
to
say
‘
f
¯
◦i
=
f
’
is
‘
f
¯
(v
s
)
=
f
(s)
for
all
s
∈
S
’.
So,
the
diagram
asserts
that
every
function
f
defined
on
the
basis
elements
extends
uniquely
to
a
linear
map
f
¯
defined
on
the
whole
of
V.
In
other
words
still,
the
function
{linear
maps
V
→
W}
f
¯
→
7→
{functions
S
→
W}
f
¯
◦
i
4
Introduction
is
bijective.
Example
0.5
Given
a
set
S
,
we
can
build
a
topological
space
D(S
)
by
equip-
ping
S
with
the
discrete
topology:
all
subsets
are
open.
With
this
topology,
any
map
from
S
to
a
space
X
is
continuous.
Again,
let
us
rephrase
this.
Define
a
function
i
:
S
→
D(S
)
by
i(s)
=
s
(s
∈
S
).
Then
D(S
)
together
with
i
has
the
following
universal
property:
S
i
/
D(S
)
∀
functions
f
∃!
continuous
f
¯
!
∀X.
In
other
words,
for
all
topological
spaces
X
and
all
functions
f
:
S
→
X,
there
exists
a
unique
continuous
map
f
¯
:
D(S
)
→
X
such
that
f
¯
◦
i
=
f
.
The
contin-
uous
map
f
¯
is
the
same
thing
as
the
function
f
,
except
that
we
are
regarding
it
as
a
continuous
map
between
topological
spaces
rather
than
a
mere
function
between
sets.
You
may
feel
that
this
universal
property
is
almost
too
trivial
to
mean
any-
thing.
But
if
we
change
the
definition
of
D(S
)
–
say
from
the
discrete
to
the
indiscrete
topology,
in
which
the
only
open
sets
are
∅
and
S
–
then
the
property
becomes
false.
So
this
property
really
does
say
something
about
the
discrete
topology.
What
it
says
is
that
all
maps
out
of
a
discrete
space
are
continuous.
Indeed,
given
S
,
the
universal
property
determines
D(S
)
and
i
uniquely
(or
rather,
uniquely
up
to
isomorphism;
but
who
could
want
more?).
The
proof
of
this
is
similar
to
that
of
Lemma
0.3
above
and
Lemma
0.7
below.
Example
0.6
Given
vector
spaces
U,
V
and
W,
a
bilinear
map
f
:
U
×
V
→
W
is
a
function
f
that
is
linear
in
each
variable:
f
(u,
v
1
+
λv
2
)
=
f
(u,
v
1
)
+
λ
f
(u,
v
2
),
f
(u
1
+
λu
2
,
v)
=
f
(u
1
,
v)
+
λ
f
(u
2
,
v)
for
all
u,
u
1
,
u
2
∈
U,
v,
v
1
,
v
2
∈
V,
and
scalars
λ.
A
good
example
is
the
scalar
product
(dot
product),
which
is
a
bilinear
map
R
n
×
R
n
(u,
v)
→
R
7→
u.v
of
real
vector
spaces.
The
vector
product
(cross
product)
R
3
×
R
3
→
R
3
is
also
bilinear.
Let
U
and
V
be
vector
spaces.
It
is
a
fact
that
there
is
a
‘universal
bilinear
5
Introduction
map
out
of
U
×
V’.
In
other
words,
there
exist
a
certain
vector
space
T
and
a
certain
bilinear
map
b
:
U
×
V
→
T
with
the
following
universal
property:
U
×
V
/
T
b
∃!
linear
f
¯
#
∀W.
∀
bilinear
f
(0.1)
Roughly
speaking,
this
property
says
that
bilinear
maps
out
of
U
×
V
corre-
spond
one-to-one
with
linear
maps
out
of
T
.
Even
without
knowing
that
such
a
T
and
b
exist,
we
can
immediately
prove
that
this
universal
property
determines
T
and
b
uniquely
up
to
isomorphism.
The
proof
is
essentially
the
same
as
that
of
Lemma
0.3,
but
looks
more
com-
plicated
because
of
the
more
complicated
universal
property.
Lemma
0.7
Let
U
and
V
be
vector
spaces.
Suppose
that
b
:
U
×
V
→
T
and
b
0
:
U
×
V
→
T
0
are
both
universal
bilinear
maps
out
of
U
×
V.
Then
T
T
0
.
More
precisely,
there
exists
a
unique
isomorphism
j
:
T
→
T
0
such
that
j
◦
b
=
b
0
.
In
the
proof
that
follows,
it
does
not
actually
matter
what
‘bilinear’,
‘linear’
or
even
‘vector
space’
mean.
The
hard
part
is
getting
the
logic
straight.
That
done,
you
should
be
able
to
see
that
there
is
really
only
one
possible
proof.
For
instance,
to
use
the
universality
of
b,
we
will
have
to
choose
some
bilinear
map
f
out
of
U
×
V.
There
are
only
two
in
sight,
b
and
b
0
,
and
we
use
each
in
the
appropriate
place.
f
b
0
Proof
In
diagram
(0.1),
take
U
×
V
−→
W
to
be
U
×
V
−→
T
0
.
This
gives
a
linear
map
j
:
T
→
T
0
satisfying
j
◦
b
=
b
0
.
Similarly,
using
the
universality
of
b
0
,
we
obtain
a
linear
map
j
0
:
T
0
→
T
satisfying
j
0
◦
b
0
=
b:
;
T
b
U
×
V
j
b
0
b
/
T
0
j
0
#
T.
Now
j
0
◦
j
:
T
→
T
is
a
linear
map
satisfying
(
j
0
◦
j)
◦
b
=
b;
but
also,
the
identity
map
1
T
:
T
→
T
is
linear
and
satisfies
1
T
◦b
=
b.
So
by
the
uniqueness
part
of
the
universal
property
of
b,
we
have
j
0
◦
j
=
1
T
.
(Here
we
took
the
‘
f
’
of
(0.1)
to
be
b.)
Similarly,
j
◦
j
0
=
1
T
0
.
So
j
is
an
isomorphism.
6
Introduction
In
Example
0.6,
it
was
stated
that
given
vector
spaces
U
and
V,
there
exists
a
pair
(T,
b)
with
the
universal
property
of
(0.1).
We
just
proved
that
there
is
essentially
only
one
such
pair
(T,
b).
The
vector
space
T
is
called
the
tensor
product
of
U
and
V,
and
is
written
as
U
⊗
V.
Tensor
products
are
very
impor-
tant
in
algebra.
They
reduce
the
study
of
bilinear
maps
to
the
study
of
linear
maps,
since
a
bilinear
map
out
of
U
×
V
is
really
the
same
thing
as
a
linear
map
out
of
U
⊗
V.
However,
tensor
products
will
not
be
important
in
this
book.
The
real
lesson
for
us
is
that
it
is
safe
to
speak
of
the
tensor
product,
not
just
a
tensor
product,
and
the
reason
for
that
is
Lemma
0.7.
This
is
a
general
point
that
applies
to
anything
satisfying
a
universal
property.
Once
you
know
a
universal
property
of
an
object,
it
often
does
no
harm
to
forget
how
it
was
constructed.
For
instance,
if
you
look
through
a
pile
of
algebra
books,
you
will
find
several
different
ways
of
constructing
the
ten-
sor
product
of
two
vector
spaces.
But
once
you
have
proved
that
the
tensor
product
satisfies
the
universal
property,
you
can
forget
the
construction.
The
universal
property
tells
you
all
you
need
to
know,
because
it
determines
the
object
uniquely
up
to
isomorphism.
Example
0.8
Let
θ
:
G
→
H
be
a
homomorphism
of
groups.
Associated
with
θ
is
a
diagram
ker(θ)
ι
/
G
θ
/
/
H,
ε
(0.2)
where
ι
is
the
inclusion
of
ker(θ)
into
G
and
ε
is
the
trivial
homomorphism.
‘Inclusion’
means
that
ι(x)
=
x
for
all
x
∈
ker(θ),
and
‘trivial’
means
that
ε(g)
=
1
for
all
g
∈
G.
The
symbol
,→
is
often
used
for
inclusions;
it
is
a
combination
of
a
subset
symbol
⊂
and
an
arrow.
The
map
ι
into
G
satisfies
θ
◦
ι
=
ε
◦
ι,
and
is
universal
as
such.
Exercise
0.11
asks
you
to
make
this
precise.
Here
is
our
final
example
of
a
universal
property.
Example
0.9
Take
a
topological
space
covered
by
two
open
subsets:
X
=
U
∪
V.
The
diagram
i
/
U
∩
V
U
_
_
j
V
i
0
/
X
j
0
of
inclusion
maps
has
a
universal
property
in
the
world
of
topological
spaces
7
Introduction
and
continuous
maps,
as
follows:
U
∩
V
_
i
j
V
i
0
/
U
_
/
X
j
0
∀
f
∃!h
∀g
+
(0.3)
∀Y.
The
diagram
means
that
given
Y,
f
and
g
such
that
f
◦
i
=
g
◦
j,
there
is
exactly
one
continuous
map
h
:
X
→
Y
such
that
h
◦
j
0
=
f
and
h
◦
i
0
=
g.
Under
favourable
conditions,
the
induced
diagram
i
∗
π
1
(U
∩
V)
j
0∗
j
∗
π
1
(V)
/
π
1
(U)
i
0∗
/
π
1
(X)
of
fundamental
groups
has
the
same
property
in
the
world
of
groups
and
group
homomorphisms.
This
is
van
Kampen’s
theorem.
In
fact,
van
Kampen
stated
his
theorem
in
a
much
more
complicated
way.
Stating
it
transparently
requires
some
categorical
language,
but
he
was
working
in
the
1930s,
before
category
theory
had
been
born.
You
have
now
seen
several
examples
of
universal
properties.
As
this
book
progresses,
we
will
develop
different
ways
of
talking
about
them.
Once
we
have
set
up
the
basic
vocabulary
of
categories
and
functors,
we
will
study
ad-
joint
functors,
then
representable
functors,
then
limits.
Each
of
these
provides
an
approach
to
universal
properties,
and
each
places
the
idea
in
a
different
light.
For
instance,
Examples
0.4
and
0.5
can
most
readily
be
described
in
terms
of
adjoint
functors,
Example
0.6
via
representable
functors,
and
Examples
0.1,
0.2,
0.8
and
0.9
in
terms
of
limits.
Exercises
0.10
Let
S
be
a
set.
The
indiscrete
topological
space
I(S
)
is
the
space
whose
set
of
points
is
S
and
whose
only
open
subsets
are
∅
and
S
itself.
Imitating
Example
0.5,
find
a
universal
property
satisfied
by
the
space
I(S
).
8
Introduction
0.11
Fix
a
group
homomorphism
θ
:
G
→
H.
Find
a
universal
property
satis-
fied
by
the
pair
(ker(θ),
ι)
of
diagram
(0.2).
(This
property
can
–
indeed,
must
–
make
reference
to
θ.)
0.12
Verify
the
universal
property
shown
in
diagram
(0.3).
0.13
Denote
by
Z[x]
the
polynomial
ring
over
Z
in
one
variable.
(a)
Prove
that
for
all
rings
R
and
all
r
∈
R,
there
exists
a
unique
ring
homo-
morphism
φ
:
Z[x]
→
R
such
that
φ(x)
=
r.
(b)
Let
A
be
a
ring
and
a
∈
A.
Suppose
that
for
all
rings
R
and
all
r
∈
R,
there
exists
a
unique
ring
homomorphism
φ
:
A
→
R
such
that
φ(a)
=
r.
Prove
that
there
is
a
unique
isomorphism
ι
:
Z[x]
→
A
such
that
ι(x)
=
a.
0.14
Let
X
and
Y
be
vector
spaces.
(a)
For
the
purposes
of
this
exercise
only,
a
cone
is
a
triple
(V,
f
1
,
f
2
)
consisting
of
a
vector
space
V,
a
linear
map
f
1
:
V
→
X,
and
a
linear
map
f
2
:
V
→
Y.
Find
a
cone
(P,
p
1
,
p
2
)
with
the
following
property:
for
all
cones
(V,
f
1
,
f
2
),
there
exists
a
unique
linear
map
f
:
V
→
P
such
that
p
1
◦
f
=
f
1
and
p
2
◦
f
=
f
2
.
(b)
Prove
that
there
is
essentially
only
one
cone
with
the
property
stated
in
(a).
That
is,
prove
that
if
(P,
p
1
,
p
2
)
and
(P
0
,
p
0
1
,
p
0
2
)
both
have
this
property
then
there
is
an
isomorphism
i
:
P
→
P
0
such
that
p
0
1
◦
i
=
p
1
and
p
0
2
◦
i
=
p
2
.
(c)
For
the
purposes
of
this
exercise
only,
a
cocone
is
a
triple
(V,
f
1
,
f
2
)
con-
sisting
of
a
vector
space
V,
a
linear
map
f
1
:
X
→
V,
and
a
linear
map
f
2
:
Y
→
V.
Find
a
cocone
(Q,
q
1
,
q
2
)
with
the
following
property:
for
all
cocones
(V,
f
1
,
f
2
),
there
exists
a
unique
linear
map
f
:
Q
→
V
such
that
f
◦
q
1
=
f
1
and
f
◦
q
2
=
f
2
.
(d)
Prove
that
there
is
essentially
only
one
cocone
with
the
property
stated
in
(c),
in
a
sense
that
you
should
make
precise.
1
Categories,
functors
and
natural
transformations
A
category
is
a
system
of
related
objects.
The
objects
do
not
live
in
isolation:
there
is
some
notion
of
map
between
objects,
binding
them
together.
Typical
examples
of
what
‘object’
might
mean
are
‘group’
and
‘topological
space’,
and
typical
examples
of
what
‘map’
might
mean
are
‘homomorphism’
and
‘continuous
map’,
respectively.
We
will
see
many
examples,
and
we
will
also
learn
that
some
categories
have
a
very
different
flavour
from
the
two
just
mentioned.
In
fact,
the
‘maps’
of
category
theory
need
not
be
anything
like
maps
in
the
sense
that
you
are
most
likely
to
be
familiar
with.
Categories
are
themselves
mathematical
objects,
and
with
that
in
mind,
it
is
unsurprising
that
there
is
a
good
notion
of
‘map
between
categories’.
Such
maps
are
called
functors.
More
surprising,
perhaps,
is
the
existence
of
a
third
level:
we
can
talk
about
maps
between
functors,
which
are
called
natural
trans-
formations.
These,
then,
are
maps
between
maps
between
categories.
In
fact,
it
was
the
desire
to
formalize
the
notion
of
natural
transformation
that
led
to
the
birth
of
category
theory.
By
the
early
1940s,
researchers
in
algebraic
topology
had
started
to
use
the
phrase
‘natural
transformation’,
but
only
in
an
informal
way.
Two
mathematicians,
Samuel
Eilenberg
and
Saunders
Mac
Lane,
saw
that
a
precise
definition
was
needed.
But
before
they
could
define
natural
transformation,
they
had
to
define
functor;
and
before
they
could
define
functor,
they
had
to
define
category.
And
so
the
subject
was
born.
Nowadays,
the
uses
of
category
theory
have
spread
far
beyond
algebraic
topology.
Its
tentacles
extend
into
most
parts
of
pure
mathematics.
They
also
reach
some
parts
of
applied
mathematics;
perhaps
most
notably,
category
the-
ory
has
become
a
standard
tool
in
certain
parts
of
computer
science.
Applied
mathematics
is
more
than
just
applied
differential
equations!
9
10
Categories,
functors
and
natural
transformations
1.1
Categories
Definition
1.1.1
A
category
A
consists
of:
•
a
collection
ob(A
)
of
objects;
•
for
each
A,
B
∈
ob(A
),
a
collection
A
(A,
B)
of
maps
or
arrows
or
mor-
phisms
from
A
to
B;
•
for
each
A,
B,
C
∈
ob(A
),
a
function
A
(B,
C)
×
A
(A,
B)
→
(g,
f
)
7→
A
(A,
C)
g
◦
f,
called
composition;
•
for
each
A
∈
ob(A
),
an
element
1
A
of
A
(A,
A),
called
the
identity
on
A,
satisfying
the
following
axioms:
•
associativity:
for
each
f
∈
A
(A,
B),
g
∈
A
(B,
C)
and
h
∈
A
(C,
D),
we
have
(h
◦
g)
◦
f
=
h
◦
(g
◦
f
);
•
identity
laws:
for
each
f
∈
A
(A,
B),
we
have
f
◦
1
A
=
f
=
1
B
◦
f
.
Remarks
1.1.2
(a)
We
often
write:
A
∈
A
f
f
:
A
→
B
or
A
−→
B
gf
to
mean
A
∈
ob(A
);
to
mean
to
mean
f
∈
A
(A,
B);
g
◦
f
.
People
also
write
A
(A,
B)
as
Hom
A
(A,
B)
or
Hom(A,
B).
The
notation
‘Hom’
stands
for
homomorphism,
from
one
of
the
earliest
examples
of
a
category.
(b)
The
definition
of
category
is
set
up
so
that
in
general,
from
each
string
f
1
f
2
f
n
A
0
−→
A
1
−→
·
·
·
−→
A
n
of
maps
in
A
,
it
is
possible
to
construct
exactly
one
map
A
0
→
A
n
(namely,
f
n
f
n−1
·
·
·
f
1
).
If
we
are
given
extra
information
then
we
may
be
able
to
construct
other
maps
A
0
→
A
n
;
for
instance,
if
we
happen
to
know
that
A
n−1
=
A
n
,
then
f
n−1
f
n−2
·
·
·
f
1
is
another
such
map.
But
we
are
speak-
ing
here
of
the
general
situation,
in
the
absence
of
extra
information.
For
example,
a
string
like
this
with
n
=
4
gives
rise
to
maps
((
f
4
f
3
)
f
2
)
f
1
A
0
(
f
4
(1
A
3
f
3
))((
f
2
f
1
)1
A
0
)
/
/
A
4
,
11
1.1
Categories
but
the
axioms
imply
that
they
are
equal.
It
is
safe
to
omit
the
brackets
and
write
both
as
f
4
f
3
f
2
f
1
.
Here
it
is
intended
that
n
≥
0.
In
the
case
n
=
0,
the
statement
is
that
for
each
object
A
0
of
a
category,
it
is
possible
to
construct
exactly
one
map
A
0
→
A
0
(namely,
the
identity
1
A
0
).
An
identity
map
can
be
thought
of
as
a
zero-fold
composite,
in
much
the
same
way
that
the
number
1
can
be
thought
of
as
the
product
of
zero
numbers.
(c)
We
often
speak
of
commutative
diagrams.
For
instance,
given
objects
and
maps
/
B
f
A
h
C
i
/
D
j
/
E
g
in
a
category,
we
say
that
the
diagram
commutes
if
g
f
=
jih.
Generally,
a
diagram
is
said
to
commute
if
whenever
there
are
two
paths
from
an
object
X
to
an
object
Y,
the
map
from
X
to
Y
obtained
by
composing
along
one
path
is
equal
to
the
map
obtained
by
composing
along
the
other.
(d)
The
slightly
vague
word
‘collection’
means
roughly
the
same
as
‘set’,
al-
though
if
you
know
about
such
things,
it
is
better
to
interpret
it
as
meaning
‘class’.
We
come
back
to
this
in
Chapter
3.
(e)
If
f
∈
A
(A,
B),
we
call
A
the
domain
and
B
the
codomain
of
f
.
Every
map
in
every
category
has
a
definite
domain
and
a
definite
codomain.
(If
you
believe
it
makes
sense
to
form
the
intersection
of
an
arbitrary
pair
of
abstract
sets,
you
should
add
to
the
definition
of
category
the
condition
that
A
(A,
B)
∩
A
(A
0
,
B
0
)
=
∅
unless
A
=
A
0
and
B
=
B
0
.)
Examples
1.1.3
(Categories
of
mathematical
structures)
(a)
There
is
a
category
Set
described
as
follows.
Its
objects
are
sets.
Given
sets
A
and
B,
a
map
from
A
to
B
in
the
category
Set
is
exactly
what
is
ordinarily
called
a
map
(or
mapping,
or
function)
from
A
to
B.
Composition
in
the
category
is
ordinary
composition
of
functions,
and
the
identity
maps
are
again
what
you
would
expect.
In
situations
such
as
this,
we
often
do
not
bother
to
specify
the
compo-
sition
and
identities.
We
write
‘the
category
of
sets
and
functions’,
leaving
the
reader
to
guess
the
rest.
In
fact,
we
usually
go
further
and
call
it
just
‘the
category
of
sets’.
(b)
There
is
a
category
Grp
of
groups,
whose
objects
are
groups
and
whose
maps
are
group
homomorphisms.
(c)
Similarly,
there
is
a
category
Ring
of
rings
and
ring
homomorphisms.
12
Categories,
functors
and
natural
transformations
(d)
For
each
field
k,
there
is
a
category
Vect
k
of
vector
spaces
over
k
and
linear
maps
between
them.
(e)
There
is
a
category
Top
of
topological
spaces
and
continuous
maps.
This
chapter
is
mostly
about
the
interaction
between
categories,
rather
than
what
goes
on
inside
them.
We
will,
however,
need
the
following
definition.
Definition
1.1.4
A
map
f
:
A
→
B
in
a
category
A
is
an
isomorphism
if
there
exists
a
map
g
:
B
→
A
in
A
such
that
g
f
=
1
A
and
f
g
=
1
B
.
In
the
situation
of
Definition
1.1.4,
we
call
g
the
inverse
of
f
and
write
g
=
f
−1
.
(The
word
‘the’
is
justified
by
Exercise
1.1.13.)
If
there
exists
an
isomorphism
from
A
to
B,
we
say
that
A
and
B
are
isomorphic
and
write
A
B.
Example
1.1.5
The
isomorphisms
in
Set
are
exactly
the
bijections.
This
statement
is
not
quite
a
logical
triviality.
It
amounts
to
the
assertion
that
a
function
has
a
two-sided
inverse
if
and
only
if
it
is
injective
and
surjective.
Example
1.1.6
The
isomorphisms
in
Grp
are
exactly
the
isomorphisms
of
groups.
Again,
this
is
not
quite
trivial,
at
least
if
you
were
taught
that
the
def-
inition
of
group
isomorphism
is
‘bijective
homomorphism’.
In
order
to
show
that
this
is
equivalent
to
being
an
isomorphism
in
Grp,
you
have
to
prove
that
the
inverse
of
a
bijective
homomorphism
is
also
a
homomorphism.
Similarly,
the
isomorphisms
in
Ring
are
exactly
the
isomorphisms
of
rings.
Example
1.1.7
The
isomorphisms
in
Top
are
exactly
the
homeomorphisms.
Note
that,
in
contrast
to
the
situation
in
Grp
and
Ring,
a
bijective
map
in
Top
is
not
necessarily
an
isomorphism.
A
classic
example
is
the
map
[0,
1)
→
t
7→
{z
∈
C
|
|z|
=
1}
e
2πit
,
which
is
a
continuous
bijection
but
not
a
homeomorphism.
The
examples
of
categories
mentioned
so
far
are
important,
but
could
give
a
false
impression.
In
each
of
them,
the
objects
of
the
category
are
sets
with
structure
(such
as
a
group
structure,
a
topology,
or,
in
the
case
of
Set,
no
struc-
ture
at
all).
The
maps
are
the
functions
preserving
the
structure,
in
the
appro-
priate
sense.
And
in
each
of
them,
there
is
a
clear
sense
of
what
the
elements
of
a
given
object
are.
However,
not
all
categories
are
like
this.
In
general,
the
objects
of
a
category
are
not
‘sets
equipped
with
extra
stuff’.
Thus,
in
a
general
category,
it
does
not
make
sense
to
talk
about
the
‘elements’
of
an
object.
(At
least,
it
does
not
make
13
1.1
Categories
sense
in
an
immediately
obvious
way;
we
return
to
this
in
Definition
4.1.25.)
Similarly,
in
a
general
category,
the
maps
need
not
be
mappings
or
functions
in
the
usual
sense.
So:
The
objects
of
a
category
need
not
be
remotely
like
sets.
The
maps
in
a
category
need
not
be
remotely
like
functions.
The
next
few
examples
illustrate
these
points.
They
also
show
that,
contrary
to
the
impression
that
might
have
been
given
so
far,
categories
need
not
be
enormous.
Some
categories
are
small,
manageable
structures
in
their
own
right,
as
we
now
see.
Examples
1.1.8
(Categories
as
mathematical
structures)
(a)
A
category
can
be
specified
by
saying
directly
what
its
objects,
maps,
composition
and
identities
are.
For
example,
there
is
a
category
∅
with
no
objects
or
maps
at
all.
There
is
a
category
1
with
one
object
and
only
the
identity
map.
It
can
be
drawn
like
this:
•
(Since
every
object
is
required
to
have
an
identity
map
on
it,
we
usually
do
not
bother
to
draw
the
identities.)
There
is
another
category
that
can
be
drawn
as
•→•
f
A
−→
B,
or
with
two
objects
and
one
non-identity
map,
from
the
first
object
to
the
second.
(Composition
is
defined
in
the
only
possible
way.)
To
reiterate
the
points
made
above,
it
is
not
obvious
what
an
‘element’
of
A
or
B
would
be,
or
how
one
could
regard
f
as
a
‘function’
of
any
sort.
It
is
easy
to
make
up
more
complicated
examples.
For
instance,
here
are
three
more
categories:
?
B
•
/
/
•
A
•
g
f
kj
gf
/
C
•
o
j
k
•
f
h
j=g
f
h
/
•
g
/
•
(b)
Some
categories
contain
no
maps
at
all
apart
from
identities
(which,
as
categories,
they
are
obliged
to
have).
These
are
called
discrete
categories.
A
discrete
category
amounts
to
just
a
class
of
objects.
More
poetically,
a
category
is
a
collection
of
objects
related
to
one
another
to
a
greater
or
lesser
degree;
a
discrete
category
is
the
extreme
case
in
which
each
object
is
totally
isolated
from
its
companions.
14
Categories,
functors
and
natural
transformations
(c)
A
group
is
essentially
the
same
thing
as
a
category
that
has
only
one
object
and
in
which
all
the
maps
are
isomorphisms.
To
understand
this,
first
consider
a
category
A
with
just
one
object.
It
is
not
important
what
letter
or
symbol
we
use
to
denote
the
object;
let
us
call
it
A.
Then
A
consists
of
a
set
(or
class)
A
(A,
A),
an
associative
composition
function
◦
:
A
(A,
A)
×
A
(A,
A)
→
A
(A,
A),
and
a
two-sided
unit
1
A
∈
A
(A,
A).
This
would
make
A
(A,
A)
into
a
group,
except
that
we
have
not
mentioned
inverses.
However,
to
say
that
every
map
in
A
is
an
isomorphism
is
exactly
to
say
that
every
element
of
A
(A,
A)
has
an
inverse
with
respect
to
◦.
If
we
write
G
for
the
group
A
(A,
A),
then
the
situation
is
this:
category
A
with
single
object
A
corresponding
group
G
maps
in
A
◦
in
A
1
A
elements
of
G
·
in
G
1
∈
G
The
category
A
looks
something
like
this:
,
A
Z
The
arrows
represent
different
maps
A
→
A,
that
is,
different
elements
of
the
group
G.
What
the
object
of
A
is
called
makes
no
difference.
It
matters
exactly
as
much
as
whether
we
choose
x
or
y
or
t
to
denote
some
variable
in
an
algebra
problem,
which
is
to
say,
not
at
all.
Later
we
will
define
‘equiv-
alence’
of
categories,
which
will
enable
us
to
make
a
precise
statement:
the
category
of
groups
is
equivalent
to
the
category
of
(small)
one-object
categories
in
which
every
map
is
an
isomorphism
(Example
3.2.11).
The
first
time
one
meets
the
idea
that
a
group
is
a
kind
of
category,
it
is
tempting
to
dismiss
it
as
a
coincidence
or
a
trick.
But
it
is
not;
there
is
real
content.
To
see
this,
suppose
that
your
education
had
been
shuffled
and
that
you
already
knew
about
categories
before
being
taught
about
groups.
In
your
first
group
theory
class,
the
lecturer
declares
that
a
group
is
supposed
to
be
the
system
of
all
symmetries
of
an
object.
A
symmetry
of
an
object
X,
she
says,
is
a
way
of
mapping
X
to
itself
in
a
reversible
or
invertible
manner.
At
this
point,
you
realize
that
she
is
talking
about
a
very
special
type
of
1.1
Categories
15
category.
In
general,
a
category
is
a
system
consisting
of
all
the
mappings
(not
usually
just
the
invertible
ones)
between
many
objects
(not
usually
just
one).
So
a
group
is
just
a
category
with
the
special
properties
that
all
the
maps
are
invertible
and
there
is
only
one
object.
(d)
The
inverses
played
no
essential
part
in
the
previous
example,
suggesting
that
it
is
worth
thinking
about
‘groups
without
inverses’.
These
are
called
monoids.
Formally,
a
monoid
is
a
set
equipped
with
an
associative
binary
opera-
tion
and
a
two-sided
unit
element.
Groups
describe
the
reversible
transfor-
mations,
or
symmetries,
that
can
be
applied
to
an
object;
monoids
describe
the
not-necessarily-reversible
transformations.
For
instance,
given
any
set
X,
there
is
a
group
consisting
of
all
bijections
X
→
X,
and
there
is
a
mo-
noid
consisting
of
all
functions
X
→
X.
In
both
cases,
the
binary
operation
is
composition
and
the
unit
is
the
identity
function
on
X.
Another
example
of
a
monoid
is
the
set
N
=
{0,
1,
2,
.
.
.}
of
natural
numbers,
with
+
as
the
operation
and
0
as
the
unit.
Alternatively,
we
could
take
the
set
N
with
·
as
the
operation
and
1
as
the
unit.
A
category
with
one
object
is
essentially
the
same
thing
as
a
monoid,
by
the
same
argument
as
for
groups.
This
is
stated
formally
in
Exam-
ple
3.2.11.
(e)
A
preorder
is
a
reflexive
transitive
binary
relation.
A
preordered
set
(S
,
≤)
is
a
set
S
together
with
a
preorder
≤
on
it.
Examples:
S
=
R
and
≤
has
its
usual
meaning;
S
is
the
set
of
subsets
of
{1,
.
.
.
,
10}
and
≤
is
⊆
(inclusion);
S
=
Z
and
a
≤
b
means
that
a
divides
b.
A
preordered
set
can
be
regarded
as
a
category
A
in
which,
for
each
A,
B
∈
A
,
there
is
at
most
one
map
from
A
to
B.
To
see
this,
consider
a
category
A
with
this
property.
It
is
not
important
what
letter
we
use
to
denote
the
unique
map
from
an
object
A
to
an
object
B;
all
we
need
to
record
is
which
pairs
(A,
B)
of
objects
have
the
property
that
a
map
A
→
B
does
exist.
Let
us
write
A
≤
B
to
mean
that
there
exists
a
map
A
→
B.
Since
A
is
a
category,
and
categories
have
composition,
if
A
≤
B
≤
C
then
A
≤
C.
Since
categories
also
have
identities,
A
≤
A
for
all
A.
The
associativity
and
identity
axioms
are
automatic.
So,
A
amounts
to
a
collection
of
objects
equipped
with
a
transitive
reflexive
binary
relation,
that
is,
a
preorder.
One
can
think
of
the
unique
map
A
→
B
as
the
statement
or
assertion
that
A
≤
B.
An
order
on
a
set
is
a
preorder
≤
with
the
property
that
if
A
≤
B
and
B
≤
A
then
A
=
B.
(Equivalently,
if
A
B
in
the
corresponding
category
then
A
=
B.)
Ordered
sets
are
also
called
partially
ordered
sets
or
posets.
16
Categories,
functors
and
natural
transformations
An
example
of
a
preorder
that
is
not
an
order
is
the
divisibility
relation
|
on
Z:
for
there
we
have
2
|
−2
and
−2
|
2
but
2
,
−2.
Here
are
two
ways
of
constructing
new
categories
from
old.
Construction
1.1.9
Every
category
A
has
an
opposite
or
dual
category
A
op
,
defined
by
reversing
the
arrows.
Formally,
ob(A
op
)
=
ob(A
)
and
A
op
(B,
A)
=
A
(A,
B)
for
all
objects
A
and
B.
Identities
in
A
op
are
the
same
as
in
A
.
Composition
in
A
op
is
the
same
as
in
A
,
but
with
the
argu-
f
g
ments
reversed.
To
spell
this
out:
if
A
−→
B
−→
C
are
maps
in
A
op
then
f
f
◦g
g
A
←−
B
←−
C
are
maps
in
A
;
these
give
rise
to
a
map
A
←−
C
in
A
,
and
the
composite
of
the
original
pair
of
maps
is
the
corresponding
map
A
→
C
in
A
op
.
So,
arrows
A
→
B
in
A
correspond
to
arrows
B
→
A
in
A
op
.
According
to
the
definition
above,
if
f
:
A
→
B
is
an
arrow
in
A
then
the
corresponding
arrow
B
→
A
in
A
op
is
also
called
f
.
Some
people
prefer
to
give
it
a
different
name,
such
as
f
op
.
Remark
1.1.10
The
principle
of
duality
is
fundamental
to
category
theory.
Informally,
it
states
that
every
categorical
definition,
theorem
and
proof
has
a
dual,
obtained
by
reversing
all
the
arrows.
Invoking
the
principle
of
du-
ality
can
save
work:
given
any
theorem,
reversing
the
arrows
throughout
its
statement
and
proof
produces
a
dual
theorem.
Numerous
examples
of
duality
appear
throughout
this
book.
Construction
1.1.11
Given
categories
A
and
B,
there
is
a
product
cate-
gory
A
×
B,
in
which
ob(A
×
B)
=
ob(A
)
×
ob(B),
(A
×
B)((A,
B),
(A
0
,
B
0
))
=
A
(A,
A
0
)
×
B(B,
B
0
).
Put
another
way,
an
object
of
the
product
category
A
×B
is
a
pair
(A,
B)
where
A
∈
A
and
B
∈
B.
A
map
(A,
B)
→
(A
0
,
B
0
)
in
A
×
B
is
a
pair
(
f,
g)
where
f
:
A
→
A
0
in
A
and
g
:
B
→
B
0
in
B.
For
the
definitions
of
composition
and
identities
in
A
×
B,
see
Exercise
1.1.14.
Exercises
1.1.12
Find
three
examples
of
categories
not
mentioned
above.
1.1.13
Show
that
a
map
in
a
category
can
have
at
most
one
inverse.
That
is,
given
a
map
f
:
A
→
B,
show
that
there
is
at
most
one
map
g
:
B
→
A
such
that
g
f
=
1
A
and
f
g
=
1
B
.
17
1.2
Functors
1.1.14
Let
A
and
B
be
categories.
Construction
1.1.11
defined
the
product
category
A
×
B,
except
that
the
definitions
of
composition
and
identities
in
A
×
B
were
not
given.
There
is
only
one
sensible
way
to
define
them;
write
it
down.
1.1.15
There
is
a
category
Toph
whose
objects
are
topological
spaces
and
whose
maps
X
→
Y
are
homotopy
classes
of
continuous
maps
from
X
to
Y.
What
do
you
need
to
know
about
homotopy
in
order
to
prove
that
Toph
is
a
category?
What
does
it
mean,
in
purely
topological
terms,
for
two
objects
of
Toph
to
be
isomorphic?
1.2
Functors
One
of
the
lessons
of
category
theory
is
that
whenever
we
meet
a
new
type
of
mathematical
object,
we
should
always
ask
whether
there
is
a
sensible
notion
of
‘map’
between
such
objects.
We
can
ask
this
about
categories
themselves.
The
answer
is
yes,
and
a
map
between
categories
is
called
a
functor.
Definition
1.2.1
of:
Let
A
and
B
be
categories.
A
functor
F
:
A
→
B
consists
•
a
function
ob(A
)
→
ob(B),
written
as
A
7→
F(A);
•
for
each
A,
A
0
∈
A
,
a
function
A
(A,
A
0
)
→
B(F(A),
F(A
0
)),
written
as
f
7→
F(
f
),
satisfying
the
following
axioms:
f
0
f
•
F(
f
0
◦
f
)
=
F(
f
0
)
◦
F(
f
)
whenever
A
−→
A
0
−→
A
00
in
A
;
•
F(1
A
)
=
1
F(A)
whenever
A
∈
A
.
Remarks
1.2.2
string
(a)
The
definition
of
functor
is
set
up
so
that
from
each
f
1
f
n
A
0
−→
·
·
·
−→
A
n
of
maps
in
A
(with
n
≥
0),
it
is
possible
to
construct
exactly
one
map
F(A
0
)
→
F(A
n
)
18
Categories,
functors
and
natural
transformations
in
B.
For
example,
given
maps
f
1
f
2
f
3
f
4
A
0
−→
A
1
−→
A
2
−→
A
3
−→
A
4
in
A
,
we
can
construct
maps
F(
f
4
f
3
)F(
f
2
f
1
)
F(A
0
)
F(1
A
4
)F(
f
4
)F(
f
3
f
2
)F(
f
1
)
/
/
F(A
4
)
in
B,
but
the
axioms
imply
that
they
are
equal.
(b)
We
are
familiar
with
the
idea
that
structures
and
the
structure-preserving
maps
between
them
form
a
category
(such
as
Grp,
Ring,
etc.).
In
particu-
lar,
this
applies
to
categories
and
functors:
there
is
a
category
CAT
whose
objects
are
categories
and
whose
maps
are
functors.
One
part
of
this
statement
is
that
functors
can
be
composed.
That
is,
F
G
G◦F
given
functors
A
−→
B
−→
C
,
there
arises
a
new
functor
A
−→
C
,
defined
in
the
obvious
way.
Another
is
that
for
every
category
A
,
there
is
an
identity
functor
1
A
:
A
→
A
.
Examples
1.2.3
Perhaps
the
easiest
examples
of
functors
are
the
so-called
forgetful
functors.
(This
is
an
informal
term,
with
no
precise
definition.)
For
instance:
(a)
There
is
a
functor
U
:
Grp
→
Set
defined
as
follows:
if
G
is
a
group
then
U(G)
is
the
underlying
set
of
G
(that
is,
its
set
of
elements),
and
if
f
:
G
→
H
is
a
group
homomorphism
then
U(
f
)
is
the
function
f
itself.
So
U
forgets
the
group
structure
of
groups
and
forgets
that
group
homomor-
phisms
are
homomorphisms.
(b)
Similarly,
there
is
a
functor
Ring
→
Set
forgetting
the
ring
structure
on
rings,
and
(for
any
field
k)
there
is
a
functor
Vect
k
→
Set
forgetting
the
vector
space
structure
on
vector
spaces.
(c)
Forgetful
functors
do
not
have
to
forget
all
the
structure.
For
example,
let
Ab
be
the
category
of
abelian
groups.
There
is
a
functor
Ring
→
Ab
that
forgets
the
multiplicative
structure,
remembering
just
the
underlying
additive
group.
Or,
let
Mon
be
the
category
of
monoids.
There
is
a
functor
U
:
Ring
→
Mon
that
forgets
the
additive
structure,
remembering
just
the
underlying
multiplicative
monoid.
(That
is,
if
R
is
a
ring
then
U(R)
is
the
set
R
made
into
a
monoid
via
·
and
1.)
(d)
There
is
an
inclusion
functor
U
:
Ab
→
Grp
defined
by
U(A)
=
A
for
any
abelian
group
A
and
U(
f
)
=
f
for
any
homomorphism
f
of
abelian
groups.
It
forgets
that
abelian
groups
are
abelian.
1.2
Functors
19
The
forgetful
functors
in
examples
(a)–(c)
forget
structure
on
the
objects,
but
that
of
example
(d)
forgets
a
property.
Nevertheless,
it
turns
out
to
be
con-
venient
to
use
the
same
word,
‘forgetful’,
in
both
situations.
Although
forgetting
is
a
trivial
operation,
there
are
situations
in
which
it
is
powerful.
For
example,
it
is
a
theorem
that
the
order
of
any
finite
field
is
a
prime
power.
An
important
step
in
the
proof
is
to
simply
forget
that
the
field
is
a
field,
remembering
only
that
it
is
a
vector
space
over
its
subfield
{0,
1,
1
+
1,
1
+
1
+
1,
.
.
.}.
Examples
1.2.4
Free
functors
are
in
some
sense
dual
to
forgetful
functors
(as
we
will
see
in
the
next
chapter),
although
they
are
less
elementary.
Again,
‘free
functor’
is
an
informal
but
useful
term.
(a)
Given
any
set
S
,
one
can
build
the
free
group
F(S
)
on
S
.
This
is
a
group
containing
S
as
a
subset
and
with
no
further
properties
other
than
those
it
is
forced
to
have,
in
a
sense
made
precise
in
Section
2.1.
Intuitively,
the
group
F(S
)
is
obtained
from
the
set
S
by
adding
just
enough
new
elements
that
it
becomes
a
group,
but
without
imposing
any
equations
other
than
those
forced
by
the
definition
of
group.
A
little
more
precisely,
the
elements
of
F(S
)
are
formal
expressions
or
words
such
as
x
−4
yx
2
zy
−3
(where
x,
y,
z
∈
S
).
Two
such
words
are
seen
as
equal
if
one
can
be
obtained
from
the
other
by
the
usual
cancellation
rules,
so
that,
for
example,
x
3
xy,
x
4
y,
and
x
2
y
−1
yx
2
y
all
represent
the
same
element
of
F(S
).
To
multiply
two
words,
just
write
one
followed
by
the
other;
for
instance,
x
−4
yx
times
xzy
−3
is
x
−4
yx
2
zy
−3
.
This
construction
assigns
to
each
set
S
a
group
F(S
).
In
fact,
F
is
a
functor:
any
map
of
sets
f
:
S
→
S
0
gives
rise
to
a
homomorphism
of
groups
F(
f
)
:
F(S
)
→
F(S
0
).
For
instance,
take
the
map
of
sets
f
:
{w,
x,
y,
z}
→
{u,
v}
defined
by
f
(w)
=
f
(x)
=
f
(y)
=
u
and
f
(z)
=
v.
This
gives
rise
to
a
homomorphism
F(
f
)
:
F({w,
x,
y,
z})
→
F({u,
v}),
which
maps
x
−4
yx
2
zy
−3
∈
F({w,
x,
y,
z})
to
u
−4
uu
2
vu
−3
=
u
−1
vu
−3
∈
F({u,
v}).
(b)
Similarly,
we
can
construct
the
free
commutative
ring
F(S
)
on
a
set
S
,
giving
a
functor
F
from
Set
to
the
category
CRing
of
commutative
rings.
In
fact,
F(S
)
is
something
familiar,
namely,
the
ring
of
polynomials
over
Z
in
commuting
variables
x
s
(s
∈
S
).
(A
polynomial
is,
after
all,
just
a
20
Categories,
functors
and
natural
transformations
formal
expression
built
from
the
variables
using
the
ring
operations
+,
−
and
·.)
For
example,
if
S
is
a
two-element
set
then
F(S
)
Z[x,
y].
(c)
We
can
also
construct
the
free
vector
space
on
a
set.
Fix
a
field
k.
The
free
functor
F
:
Set
→
Vect
k
is
defined
on
objects
by
taking
F(S
)
to
be
a
vector
space
with
basis
S
.
Any
two
such
vector
spaces
are
isomorphic;
but
it
is
perhaps
not
obvious
that
there
is
any
such
vector
space
at
all,
so
we
have
to
construct
one.
Loosely,
F(S
)
is
the
set
of
all
formal
k-linear
combinations
of
elements
of
S
,
that
is,
expressions
X
λ
s
s
s∈S
where
each
λ
s
is
a
scalar
and
there
are
only
finitely
many
values
of
s
such
that
λ
s
,
0.
(This
restriction
is
imposed
because
one
can
only
take
finite
sums
in
a
vector
space.)
Elements
of
F(S
)
can
be
added:
X
X
X
(λ
s
+
µ
s
)s.
µ
s
s
=
λ
s
s
+
s∈S
s∈S
s∈S
There
is
also
a
scalar
multiplication
on
F(S
):
X
X
(cλ
s
)s
λ
s
s
=
c
·
s∈S
s∈S
(c
∈
k).
In
this
way,
F(S
)
becomes
a
vector
space.
To
be
completely
precise
and
avoid
talking
about
‘expressions’,
we
can
define
F(S
)
to
be
the
set
of
all
functions
λ
:
S
→
k
such
that
{s
∈
S
|
λ(s)
,
0}
is
finite.
(Think
of
such
a
function
λ
as
corresponding
to
the
P
expression
s∈S
λ(s)s.)
To
define
addition
on
F(S
),
we
must
define
for
each
λ,
µ
∈
F(S
)
a
sum
λ
+
µ
∈
F(S
);
it
is
given
by
(λ
+
µ)(s)
=
λ(s)
+
µ(s)
(s
∈
S
).
Similarly,
the
scalar
multiplication
is
given
by
(c
·
λ)(s)
=
c
·
λ(s)
(c
∈
k,
λ
∈
F(S
),
s
∈
S
).
Rings
and
vector
spaces
have
the
special
property
that
it
is
relatively
easy
to
write
down
an
explicit
formula
for
the
free
functor.
The
case
of
groups
is
much
more
typical.
For
most
types
of
algebraic
structure,
describing
the
free
functor
requires
as
much
fussy
work
as
it
does
for
groups.
We
return
to
this
point
in
Example
2.1.3
and
Example
6.3.11
(where
we
see
how
to
avoid
the
fussy
work
entirely).
Examples
1.2.5
(Functors
in
algebraic
topology)
Historically,
some
of
the
first
examples
of
functors
arose
in
algebraic
topology.
There,
the
strategy
is
21
1.2
Functors
to
learn
about
a
space
by
extracting
data
from
it
in
some
clever
way,
assem-
bling
that
data
into
an
algebraic
structure,
then
studying
the
algebraic
structure
instead
of
the
original
space.
Algebraic
topology
therefore
involves
many
func-
tors
from
categories
of
spaces
to
categories
of
algebras.
(a)
Let
Top
∗
be
the
category
of
topological
spaces
equipped
with
a
basepoint,
together
with
the
continuous
basepoint-preserving
maps.
There
is
a
func-
tor
π
1
:
Top
∗
→
Grp
assigning
to
each
space
X
with
basepoint
x
the
fun-
damental
group
π
1
(X,
x)
of
X
at
x.
(Some
texts
use
the
simpler
notation
π
1
(X),
ignoring
the
choice
of
basepoint.
This
is
more
or
less
safe
if
X
is
path-connected,
but
strictly
speaking,
the
basepoint
should
always
be
spec-
ified.)
That
π
1
is
a
functor
means
that
it
not
only
assigns
to
each
space-with-
basepoint
(X,
x)
a
group
π
1
(X,
x),
but
also
assigns
to
each
basepoint-pre-
serving
continuous
map
f
:
(X,
x)
→
(Y,
y)
a
homomorphism
π
1
(
f
)
:
π
1
(X,
x)
→
π
1
(Y,
y).
Usually
π
1
(
f
)
is
written
as
f
∗
.
The
functoriality
axioms
say
that
(g
◦
f
)
∗
=
g
∗
◦
f
∗
and
(1
(X,x)
)
∗
=
1
π
1
(X,x)
.
(b)
For
each
n
∈
N,
there
is
a
functor
H
n
:
Top
→
Ab
assigning
to
a
space
its
nth
homology
group
(in
any
of
several
possible
senses).
Example
1.2.6
Any
system
of
polynomial
equations
such
as
2x
2
+
y
2
−
3z
2
=
1
(1.1)
x
+
x
=
y
(1.2)
3
2
gives
rise
to
a
functor
CRing
→
Set.
Indeed,
for
each
commutative
ring
A,
let
F(A)
be
the
set
of
triples
(x,
y,
z)
∈
A
×
A
×
A
satisfying
equations
(1.1)
and
(1.2).
Whenever
f
:
A
→
B
is
a
ring
homomorphism
and
(x,
y,
z)
∈
F(A),
we
have
(
f
(x),
f
(y),
f
(z))
∈
F(B);
so
the
map
of
rings
f
:
A
→
B
induces
a
map
of
sets
F(
f
)
:
F(A)
→
F(B).
This
defines
a
functor
F
:
CRing
→
Set.
In
algebraic
geometry,
a
scheme
is
a
functor
CRing
→
Set
with
certain
properties.
(This
is
not
the
most
common
way
of
phrasing
the
definition,
but
it
is
equivalent.)
The
functor
F
above
is
a
simple
example.
Example
1.2.7
Let
G
and
H
be
monoids
(or
groups,
if
you
prefer),
regarded
as
one-object
categories
G
and
H
.
A
functor
F
:
G
→
H
must
send
the
unique
object
of
G
to
the
unique
object
of
H
,
so
it
is
determined
by
its
effect
22
Categories,
functors
and
natural
transformations
on
maps.
Hence,
the
functor
F
:
G
→
H
amounts
to
a
function
F
:
G
→
H
such
that
F(g
0
g)
=
F(g
0
)F(g)
for
all
g
0
,
g
∈
G,
and
F(1)
=
1.
In
other
words,
a
functor
G
→
H
is
just
a
homomorphism
G
→
H.
Example
1.2.8
Let
G
be
a
monoid,
regarded
as
a
one-object
category
G
.
A
functor
F
:
G
→
Set
consists
of
a
set
S
(the
value
of
F
at
the
unique
object
of
G
)
together
with,
for
each
g
∈
G,
a
function
F(g)
:
S
→
S
,
satisfying
the
functoriality
axioms.
Writing
(F(g))(s)
=
g
·
s,
we
see
that
the
functor
F
amounts
to
a
set
S
together
with
a
function
G
×
S
(g,
s)
→
S
7
→
g
·
s
satisfying
(g
0
g)
·
s
=
g
0
·
(g
·
s)
and
1
·
s
=
s
for
all
g,
g
0
∈
G
and
s
∈
S
.
In
other
words,
a
functor
G
→
Set
is
a
set
equipped
with
a
left
action
by
G:
a
left
G-set,
for
short.
Similarly,
a
functor
G
→
Vect
k
is
exactly
a
k-linear
representation
of
G,
in
the
sense
of
representation
theory.
This
can
reasonably
be
taken
as
the
defini-
tion
of
representation.
Example
1.2.9
When
A
and
B
are
(pre)ordered
sets,
a
functor
between
the
corresponding
categories
is
exactly
an
order-preserving
map,
that
is,
a
func-
tion
f
:
A
→
B
such
that
a
≤
a
0
=⇒
f
(a)
≤
f
(a
0
).
Exercise
1.2.22
asks
you
to
verify
this.
Sometimes
we
meet
functor-like
operations
that
reverse
the
arrows,
with
a
map
A
→
A
0
in
A
giving
rise
to
a
map
F(A)
←
F(A
0
)
in
B.
Such
operations
are
called
contravariant
functors.
Definition
1.2.10
Let
A
and
B
be
categories.
A
contravariant
functor
from
A
to
B
is
a
functor
A
op
→
B.
To
avoid
confusion,
we
write
‘a
contravariant
functor
from
A
to
B’
rather
than
‘a
contravariant
functor
A
→
B’.
Functors
C
→
D
correspond
one-to-one
with
functors
C
op
→
D
op
,
and
(A
op
)
op
=
A
,
so
a
contravariant
functor
from
A
to
B
can
also
be
described
as
a
functor
A
→
B
op
.
Which
description
we
use
is
not
enormously
important,
but
in
the
long
run,
the
convention
in
Definition
1.2.10
makes
life
easier.
An
ordinary
functor
A
→
B
is
sometimes
called
a
covariant
functor
from
A
to
B,
for
emphasis.
Example
1.2.11
We
can
tell
a
lot
about
a
space
by
examining
the
functions
on
it.
The
importance
of
this
principle
in
twentieth-
and
twenty-first-century
mathematics
can
hardly
be
exaggerated.
1.2
Functors
23
For
example,
given
a
topological
space
X,
let
C(X)
be
the
ring
of
continuous
real-valued
functions
on
X.
The
ring
operations
are
defined
‘pointwise’:
for
instance,
if
p
1
,
p
2
:
X
→
R
are
continuous
maps
then
the
map
p
1
+
p
2
:
X
→
R
is
defined
by
(p
1
+
p
2
)(x)
=
p
1
(x)
+
p
2
(x)
(x
∈
X).
A
continuous
map
f
:
X
→
Y
induces
a
ring
homomorphism
C(
f
)
:
C(Y)
→
C(X),
defined
at
q
∈
C(Y)
by
taking
(C(
f
))(q)
to
be
the
composite
map
f
q
X
−→
Y
−→
R.
Note
that
C(
f
)
goes
in
the
opposite
direction
from
f
.
After
checking
some
axioms
(Exercise
1.2.26),
we
conclude
that
C
is
a
contravariant
functor
from
Top
to
Ring.
While
this
particular
example
will
not
play
a
large
part
in
this
text,
it
is
worth
close
attention.
It
illustrates
the
important
idea
of
a
structure
whose
elements
are
maps
(in
this
case,
a
ring
whose
elements
are
continuous
functions).
The
way
in
which
C
becomes
a
functor,
via
composition,
is
also
important.
Similar
constructions
will
be
crucial
in
later
chapters.
For
certain
classes
of
space,
the
passage
from
X
to
C(X)
loses
no
informa-
tion:
there
is
a
way
of
reconstructing
the
space
X
from
the
ring
C(X).
For
this
and
related
reasons,
it
is
sometimes
said
that
‘algebra
is
dual
to
geometry’.
Example
1.2.12
Let
k
be
a
field.
For
any
two
vector
spaces
V
and
W
over
k,
there
is
a
vector
space
Hom(V,
W)
=
{linear
maps
V
→
W}.
The
elements
of
this
vector
space
are
themselves
maps,
and
the
vector
space
operations
(addition
and
scalar
multiplication)
are
defined
pointwise,
as
in
the
last
example.
Now
fix
a
vector
space
W.
Any
linear
map
f
:
V
→
V
0
induces
a
linear
map
f
∗
:
Hom(V
0
,
W)
→
Hom(V,
W),
defined
at
q
∈
Hom(V
0
,
W)
by
taking
f
∗
(q)
to
be
the
composite
map
f
q
V
−→
V
0
−→
W.
This
defines
a
functor
op
Hom(−,
W)
:
Vect
k
→
Vect
k
.
24
Categories,
functors
and
natural
transformations
The
symbol
‘−’
is
a
blank
or
placeholder,
into
which
arguments
can
be
in-
serted.
Thus,
the
value
of
Hom(−,
W)
at
V
is
Hom(V,
W).
Sometimes
we
use
a
blank
space
instead
of
−,
as
in
Hom(
,
W).
An
important
special
case
is
where
W
is
k,
seen
as
a
one-dimensional
vector
space
over
itself.
The
vector
space
Hom(V,
k)
is
called
the
dual
of
V,
and
is
written
as
V
∗
.
So
there
is
a
contravariant
functor
op
(
)
∗
=
Hom(−,
k)
:
Vect
k
→
Vect
k
sending
each
vector
space
to
its
dual.
Example
1.2.13
For
each
n
∈
N,
there
is
a
functor
H
n
:
Top
op
→
Ab
assign-
ing
to
a
space
its
nth
cohomology
group.
Example
1.2.14
Let
G
be
a
monoid,
regarded
as
a
one-object
category
G
.
A
functor
G
op
→
Set
is
a
right
G-set,
for
essentially
the
same
reasons
as
in
Example
1.2.8.
That
left
actions
are
covariant
functors
and
right
actions
are
contravariant
functors
is
a
consequence
of
a
basic
notational
choice:
we
write
the
value
of
a
function
f
at
an
element
x
as
f
(x),
not
(x)
f
.
Contravariant
functors
whose
codomain
is
Set
are
important
enough
to
have
their
own
special
name.
Definition
1.2.15
Set.
Let
A
be
a
category.
A
presheaf
on
A
is
a
functor
A
op
→
The
name
comes
from
the
following
special
case.
Let
X
be
a
topological
space.
Write
O(X)
for
the
poset
of
open
subsets
of
X,
ordered
by
inclusion.
View
O(X)
as
a
category,
as
in
Example
1.1.8(e).
Thus,
the
objects
of
O(X)
are
the
open
subsets
of
X,
and
for
U,
U
0
∈
O(X),
there
is
one
map
U
→
U
0
if
U
⊆
U
0
,
and
there
are
none
otherwise.
A
presheaf
on
the
space
X
is
a
presheaf
on
the
category
O(X).
For
example,
given
any
space
X,
there
is
a
presheaf
F
on
X
defined
by
F(U)
=
{continuous
functions
U
→
R}
(U
∈
O(X))
and,
whenever
U
⊆
U
0
are
open
subsets
of
X,
by
taking
the
map
F(U
0
)
→
F(U)
to
be
restriction.
Presheaves,
and
a
certain
class
of
presheaves
called
sheaves,
play
an
important
role
in
modern
geometry.
We
know
very
well
that
for
functions
between
sets,
it
is
sometimes
useful
to
consider
special
kinds
of
function
such
as
injections,
surjections
and
bijections.
We
also
know
that
the
notions
of
injection
and
subset
are
related:
for
instance,
25
1.2
Functors
A
A
F(A)
F
−→
A
0
g
B
F(A
0
)
Figure
1.1
Fullness
and
faithfulness.
whenever
B
is
a
subset
of
A,
there
is
an
injection
B
→
A
given
by
inclusion.
In
this
section
and
the
next,
we
introduce
some
similar
notions
for
functors
between
categories,
beginning
with
the
following
definitions.
Definition
1.2.16
A
functor
F
:
A
→
B
is
faithful
(respectively,
full)
if
for
each
A,
A
0
∈
A
,
the
function
A
(A,
A
0
)
f
→
B(F(A),
F(A
0
))
7
→
F(
f
)
is
injective
(respectively,
surjective).
Warning
1.2.17
Note
the
roles
of
A
and
A
0
in
the
definition.
Faithfulness
does
not
say
that
if
f
1
and
f
2
are
distinct
maps
in
A
then
F(
f
1
)
,
F(
f
2
)
(Exercise
1.2.27).
In
the
situation
of
Figure
1.1,
F
is
faithful
if
for
each
A,
A
0
and
g
as
shown,
there
is
at
most
one
dotted
arrow
that
F
sends
to
g.
It
is
full
if
for
each
such
A,
A
0
and
g,
there
is
at
least
one
dotted
arrow
that
F
sends
to
g.
Definition
1.2.18
Let
A
be
a
category.
A
subcategory
S
of
A
consists
of
a
subclass
ob(S
)
of
ob(A
)
together
with,
for
each
S
,
S
0
∈
ob(S
),
a
subclass
S
(S
,
S
0
)
of
A
(S
,
S
0
),
such
that
S
is
closed
under
composition
and
identities.
It
is
a
full
subcategory
if
S
(S
,
S
0
)
=
A
(S
,
S
0
)
for
all
S
,
S
0
∈
ob(S
).
A
full
subcategory
therefore
consists
of
a
selection
of
the
objects,
with
all
of
the
maps
between
them.
So,
a
full
subcategory
can
be
specified
simply
by
saying
what
its
objects
are.
For
example,
Ab
is
the
full
subcategory
of
Grp
consisting
of
the
groups
that
are
abelian.
Whenever
S
is
a
subcategory
of
a
category
A
,
there
is
an
inclusion
functor
I
:
S
→
A
defined
by
I(S
)
=
S
and
I(
f
)
=
f
.
It
is
automatically
faithful,
and
it
is
full
if
and
only
if
S
is
a
full
subcategory.
Warning
1.2.19
The
image
of
a
functor
need
not
be
a
subcategory.
For
ex-
26
Categories,
functors
and
natural
transformations
ample,
consider
the
functor
A
f
/
B
B
0
g
/
C
F
−→
X
?
Y
p
q
qp
/
Z
defined
by
F(A)
=
X,
F(B)
=
F(B
0
)
=
Y,
F(C)
=
Z,
F(
f
)
=
p,
and
F(g)
=
q.
Then
p
and
q
are
in
the
image
of
F,
but
qp
is
not.
Exercises
1.2.20
Find
three
examples
of
functors
not
mentioned
above.
1.2.21
Show
that
functors
preserve
isomorphism.
That
is,
prove
that
if
F
:
A
→
B
is
a
functor
and
A,
A
0
∈
A
with
A
A
0
,
then
F(A)
F(A
0
).
1.2.22
Prove
the
assertion
made
in
Example
1.2.9.
In
other
words,
given
or-
dered
sets
A
and
B,
and
denoting
by
A
and
B
the
corresponding
categories,
show
that
a
functor
A
→
B
amounts
to
an
order-preserving
map
A
→
B.
1.2.23
Two
categories
A
and
B
are
isomorphic,
written
as
A
B,
if
they
are
isomorphic
as
objects
of
CAT.
(a)
Let
G
be
a
group,
regarded
as
a
one-object
category
all
of
whose
maps
are
isomorphisms.
Then
its
opposite
G
op
is
also
a
one-object
category
all
of
whose
maps
are
isomorphisms,
and
can
therefore
be
regarded
as
a
group
too.
What
is
G
op
,
in
purely
group-theoretic
terms?
Prove
that
G
is
isomor-
phic
to
G
op
.
(b)
Find
a
monoid
not
isomorphic
to
its
opposite.
1.2.24
Is
there
a
functor
Z
:
Grp
→
Grp
with
the
property
that
Z(G)
is
the
centre
of
G
for
all
groups
G?
1.2.25
Sometimes
we
meet
functors
whose
domain
is
a
product
A
×
B
of
categories.
Here
you
will
show
that
such
a
functor
can
be
regarded
as
an
inter-
locking
pair
of
families
of
functors,
one
defined
on
A
and
the
other
defined
on
B.
(This
is
very
like
the
situation
for
bilinear
and
linear
maps.)
(a)
Let
F
:
A
×
B
→
C
be
a
functor.
Prove
that
for
each
A
∈
A
,
there
is
a
functor
F
A
:
B
→
C
defined
on
objects
B
∈
B
by
F
A
(B)
=
F(A,
B)
and
on
maps
g
in
B
by
F
A
(g)
=
F(1
A
,
g).
Prove
that
for
each
B
∈
B,
there
is
a
functor
F
B
:
A
→
C
defined
similarly.
1.3
Natural
transformations
27
(b)
Let
F
:
A
×
B
→
C
be
a
functor.
With
notation
as
in
(a),
show
that
the
families
of
functors
(F
A
)
A∈A
and
(F
B
)
B∈B
satisfy
the
following
two
conditions:
•
if
A
∈
A
and
B
∈
B
then
F
A
(B)
=
F
B
(A);
0
•
if
f
:
A
→
A
0
in
A
and
g
:
B
→
B
0
in
B
then
F
A
(g)
◦
F
B
(
f
)
=
F
B
0
(
f
)
◦
F
A
(g).
(c)
Now
take
categories
A
,
B
and
C
,
and
take
families
of
functors
(F
A
)
A∈A
and
(F
B
)
B∈B
satisfying
the
two
conditions
in
(b).
Prove
that
there
is
a
unique
functor
F
:
A
×
B
→
C
satisfying
the
equations
in
(a).
(‘There
is
a
unique
functor’
means
in
particular
that
there
is
a
functor,
so
you
have
to
prove
existence
as
well
as
uniqueness.)
1.2.26
Fill
in
the
details
of
Example
1.2.11,
thus
constructing
a
functor
C
:
Top
op
→
Ring.
1.2.27
Find
an
example
of
a
functor
F
:
A
→
B
such
that
F
is
faithful
but
there
exist
distinct
maps
f
1
and
f
2
in
A
with
F(
f
1
)
=
F(
f
2
).
1.2.28
(a)
Of
the
examples
of
functors
appearing
in
this
section,
which
are
faithful
and
which
are
full?
(b)
Write
down
one
example
of
a
functor
that
is
both
full
and
faithful,
one
that
is
full
but
not
faithful,
one
that
is
faithful
but
not
full,
and
one
that
is
neither.
1.2.29
(a)
What
are
the
subcategories
of
an
ordered
set?
Which
are
full?
(b)
What
are
the
subcategories
of
a
group?
(Careful!)
Which
are
full?
1.3
Natural
transformations
We
now
know
about
categories.
We
also
know
about
functors,
which
are
maps
between
categories.
Perhaps
surprisingly,
there
is
a
further
notion
of
‘map
be-
tween
functors’.
Such
maps
are
called
natural
transformations.
This
notion
only
applies
when
the
functors
have
the
same
domain
and
codomain:
A
F
G
/
/
B
.
To
see
how
this
might
work,
let
us
consider
a
special
case.
Let
A
be
the
discrete
category
(Example
1.1.8(b))
whose
objects
are
the
natural
numbers
0,
1,
2,
.
.
.
.
A
functor
F
from
A
to
another
category
B
is
simply
a
sequence
28
Categories,
functors
and
natural
transformations
(F
0
,
F
1
,
F
2
,
.
.
.)
of
objects
of
B.
Let
G
be
another
functor
from
A
to
B,
con-
sisting
of
another
sequence
(G
0
,
G
1
,
G
2
,
.
.
.)
of
objects
of
B.
It
would
be
rea-
sonable
to
define
a
‘map’
from
F
to
G
to
be
a
sequence
α
0
α
1
α
2
F
0
−→
G
0
,
F
1
−→
G
1
,
F
1
−→
G
2
,
.
.
.
of
maps
in
B.
The
situation
can
be
depicted
as
follows:
F
0
A
0
1
2
α
0
···
G
0
F
1
α
1
G
1
F
2
α
2
G
2
B
···
(The
right-hand
diagram
should
not
be
understood
too
literally.
Some
of
the
objects
F
i
or
G
i
might
be
equal,
and
there
might
be
much
else
in
B
besides
what
is
shown.)
This
suggests
that
in
the
general
case,
a
natural
transformation
between
F
/
/
B
should
consist
of
maps
α
:
F(A)
→
G(A),
one
for
each
functors
A
A
G
A
∈
A
.
In
the
example
above,
the
category
A
had
the
special
property
of
not
containing
any
nontrivial
maps.
In
general,
we
demand
some
kind
of
compati-
bility
between
the
maps
in
A
and
the
maps
α
A
.
F
/
/
B
be
functors.
Let
A
and
B
be
categories
and
let
A
G
α
A
A
natural
transformation
α
:
F
→
G
is
a
family
F(A)
−→
G(A)
of
Definition
1.3.1
A∈A
f
maps
in
B
such
that
for
every
map
A
−→
A
0
in
A
,
the
square
F(A)
F(
f
)
α
A
0
α
A
G(A)
/
F(A
0
)
G(
f
)
/
G(A
0
)
(1.3)
commutes.
The
maps
α
A
are
called
the
components
of
α.
Remarks
1.3.2
(a)
The
definition
of
natural
transformation
is
set
up
so
that
f
from
each
map
A
−→
A
0
in
A
,
it
is
possible
to
construct
exactly
one
map
F(A)
→
G(A
0
)
in
B.
When
f
=
1
A
,
this
map
is
α
A
.
For
a
general
f
,
it
is
the
diagonal
of
the
square
(1.3),
and
‘exactly
one’
implies
that
the
square
commutes.
1.3
Natural
transformations
29
(b)
We
write
F
A
α
&
8
B
G
to
mean
that
α
is
a
natural
transformation
from
F
to
G.
Example
1.3.3
Let
A
be
a
discrete
category,
and
let
F,
G
:
A
→
B
be
func-
tors.
Then
F
and
G
are
just
families
(F(A))
A∈A
and
(G(A))
objects
A∈A
α
of
of
A
B.
A
natural
transformation
α
:
F
→
G
is
just
a
family
F(A)
−→
G(A)
A∈A
of
maps
in
B,
as
claimed
above
in
the
case
ob
A
=
N.
In
principle,
this
fam-
ily
must
satisfy
the
naturality
axiom
(1.3)
for
every
map
f
in
A
;
but
the
only
maps
in
A
are
the
identities,
and
when
f
is
an
identity,
this
axiom
holds
auto-
matically.
Example
1.3.4
Recall
from
Examples
1.1.8
that
a
group
(or
more
generally,
a
monoid)
G
can
be
regarded
as
a
one-object
category.
Also
recall
from
Exam-
ple
1.2.8
that
a
functor
from
the
category
G
to
Set
is
nothing
but
a
left
G-set.
(Previously
we
used
G
to
denote
the
category
corresponding
to
the
group
G;
from
now
on
we
use
G
to
denote
them
both.)
Take
two
G-sets,
S
and
T
.
Since
S
and
T
can
be
regarded
as
functors
G
→
Set,
we
can
ask:
what
is
a
natural
transformation
(
α
6
Set,
S
G
T
in
concrete
terms?
Such
a
natural
transformation
consists
of
a
single
map
in
Set
(since
G
has
just
one
object),
satisfying
some
axioms.
Precisely,
it
is
a
function
α
:
S
→
T
such
that
α(g
·
s)
=
g
·
α(s)
for
all
s
∈
S
and
g
∈
G.
(Why?)
In
other
words,
it
is
just
a
map
of
G-sets,
sometimes
called
a
G-equivariant
map.
Example
1.3.5
Fix
a
natural
number
n.
In
this
example,
we
will
see
how
‘determinant
of
an
n
×
n
matrix’
can
be
understood
as
a
natural
transformation.
For
any
commutative
ring
R,
the
n
×
n
matrices
with
entries
in
R
form
a
monoid
M
n
(R)
under
multiplication.
Moreover,
any
ring
homomorphism
R
→
S
induces
a
monoid
homomorphism
M
n
(R)
→
M
n
(S
).
This
defines
a
functor
M
n
:
CRing
→
Mon
from
the
category
of
commutative
rings
to
the
category
of
monoids.
Also,
the
elements
of
any
ring
R
form
a
monoid
U(R)
under
multiplication,
giving
another
functor
U
:
CRing
→
Mon.
30
Categories,
functors
and
natural
transformations
Now,
every
n
×
n
matrix
X
over
a
commutative
ring
R
has
a
determinant
det
R
(X),
which
is
an
element
of
R.
Familiar
properties
of
determinant
–
det
R
(XY)
=
det
R
(X)det
R
(Y),
det
R
(I)
=
1
–
tell
us
that
for
each
R,
the
function
det
R
:
M
n
(R)
→
U(R)
is
a
monoid
homo-
morphism.
So,
we
have
a
family
of
maps
det
R
M
n
(R)
−→
U(R)
,
R∈CRing
and
it
makes
sense
to
ask
whether
they
define
a
natural
transformation
M
n
CRing
+
3
Mon.
det
U
Indeed,
they
do.
That
the
naturality
squares
commute
(check!)
reflects
the
fact
that
determinant
is
defined
in
the
same
way
for
all
rings.
We
do
not
use
one
definition
of
determinant
for
one
ring
and
a
different
definition
for
another
ring.
Generally
speaking,
the
naturality
axiom
(1.3)
is
supposed
to
capture
the
idea
that
the
family
(α
A
)
A∈A
is
defined
in
a
uniform
way
across
all
A
∈
A
.
Construction
1.3.6
Natural
transformations
are
a
kind
of
map,
so
we
would
expect
to
be
able
to
compose
them.
We
can.
Given
natural
transformations
F
A
α
β
/
B
,
B
G
H
there
is
a
composite
natural
transformation
F
A
(
6
B
β◦α
H
defined
by
(β
◦
α)
A
=
β
A
◦
α
A
for
all
A
∈
A
.
There
is
also
an
identity
natural
transformation
F
A
1
F
(
6
B
F
on
any
functor
F,
defined
by
(1
F
)
A
=
1
F(A)
.
So
for
any
two
categories
A
and
B,
there
is
a
category
whose
objects
are
the
functors
from
A
to
B
and
whose
maps
are
the
natural
transformations
between
them.
This
is
called
the
functor
category
from
A
to
B,
and
written
as
[A
,
B]
or
B
A
.
1.3
Natural
transformations
31
Example
1.3.7
Let
2
be
the
discrete
category
with
two
objects.
A
functor
from
2
to
a
category
B
is
a
pair
of
objects
of
B,
and
a
natural
transformation
is
a
pair
of
maps.
The
functor
category
[2,
B]
is
therefore
isomorphic
to
the
product
category
B
×
B
(Construction
1.1.11).
This
fits
well
with
the
alterna-
tive
notation
B
2
for
the
functor
category.
Example
1.3.8
Let
G
be
a
monoid.
Then
[G,
Set]
is
the
category
of
left
G-
sets,
and
[G
op
,
Set]
is
the
category
of
right
G-sets
(Example
1.2.14).
Example
1.3.9
Take
ordered
sets
A
and
B,
viewed
as
categories
(as
in
Exam-
f
ple
1.1.8(e)).
Given
order-preserving
maps
A
g
/
/
B,
viewed
as
functors
(as
in
Example
1.2.9),
there
is
at
most
one
natural
transformation
f
A
&
8
B,
g
and
there
is
one
if
and
only
if
f
(a)
≤
g(a)
for
all
a
∈
A.
(The
naturality
axiom
(1.3)
holds
automatically,
because
in
an
ordered
set,
all
diagrams
com-
mute.)
So
[A,
B]
is
an
ordered
set
too;
its
elements
are
the
order-preserving
maps
from
A
to
B,
and
f
≤
g
if
and
only
if
f
(a)
≤
g(a)
for
all
a
∈
A.
Everyday
phrases
such
as
‘the
cyclic
group
of
order
6’
and
‘the
product
of
two
spaces’
reflect
the
fact
that
given
two
isomorphic
objects
of
a
category,
we
usually
neither
know
nor
care
whether
they
are
actually
equal.
This
is
enor-
mously
important.
In
particular,
the
lesson
applies
when
the
category
concerned
is
a
functor
category.
In
other
words,
given
two
functors
F,
G
:
A
→
B,
we
usually
do
not
care
whether
they
are
literally
equal.
(Equality
would
imply
that
the
objects
F(A)
and
G(A)
of
B
were
equal
for
all
A
∈
A
,
a
level
of
detail
in
which
we
have
just
declared
ourselves
to
be
uninterested.)
What
really
matters
is
whether
they
are
naturally
isomorphic.
Definition
1.3.10
Let
A
and
B
be
categories.
A
natural
isomorphism
be-
tween
functors
from
A
to
B
is
an
isomorphism
in
[A
,
B].
An
equivalent
form
of
the
definition
is
often
useful:
F
Lemma
1.3.11
Let
A
G
α
$
B
;
be
a
natural
transformation.
Then
α
is
a
nat-
ural
isomorphism
if
and
only
if
α
A
:
F(A)
→
G(A)
is
an
isomorphism
for
all
A
∈
A
.
32
Proof
Categories,
functors
and
natural
transformations
Exercise
1.3.26.
Of
course,
we
say
that
functors
F
and
G
are
naturally
isomorphic
if
there
exists
a
natural
isomorphism
from
F
to
G.
Since
natural
isomorphism
is
just
isomorphism
in
a
particular
category
(namely,
[A
,
B]),
we
already
have
nota-
tion
for
this:
F
G.
Definition
1.3.12
F
Given
functors
A
G
/
/
B
,
we
say
that
F(A)
G(A)
naturally
in
A
if
F
and
G
are
naturally
isomorphic.
This
alternative
terminology
can
be
understood
as
follows.
If
F(A)
G(A)
naturally
in
A
then
certainly
F(A)
G(A)
for
each
individual
A,
but
more
is
true:
we
can
choose
isomorphisms
α
A
:
F(A)
→
G(A)
in
such
a
way
that
the
naturality
axiom
(1.3)
is
satisfied.
Example
1.3.13
Let
F,
G
:
A
→
B
be
functors
from
a
discrete
category
A
to
a
category
B.
Then
F
G
if
and
only
if
F(A)
G(A)
for
all
A
∈
A
.
So
in
this
case,
F(A)
G(A)
naturally
in
A
if
and
only
if
F(A)
G(A)
for
all
A.
But
this
is
only
true
because
A
is
discrete.
In
general,
it
is
emphatically
F
/
/
B
such
false.
There
are
many
examples
of
categories
and
functors
A
G
that
F(A)
G(A)
for
all
A
∈
A
,
but
not
naturally
in
A.
Exercise
1.3.31
gives
an
example
from
combinatorics.
Example
1.3.14
Let
FDVect
be
the
category
of
finite-dimensional
vector
spaces
over
some
field
k.
The
dual
vector
space
construction
defines
a
con-
travariant
functor
from
FDVect
to
itself
(Example
1.2.12),
and
the
double
dual
construction
therefore
defines
a
covariant
functor
from
FDVect
to
itself.
Moreover,
we
have
for
each
V
∈
FDVect
a
canonical
isomorphism
α
V
:
V
→
V
∗∗
.
Given
v
∈
V,
the
element
α
V
(v)
of
V
∗∗
is
‘evaluation
at
v’;
that
is,
α
V
(v)
:
V
∗
→
k
maps
φ
∈
V
∗
to
φ(v)
∈
k.
That
α
V
is
an
isomorphism
is
a
standard
result
in
the
theory
of
finite-dimensional
vector
spaces.
This
defines
a
natural
transformation
1
FDVect
FDVect
α
'
FDVect
9
(
)
∗∗
from
the
identity
functor
to
the
double
dual
functor.
By
Lemma
1.3.11,
α
is
33
1.3
Natural
transformations
a
natural
isomorphism.
So
1
FDVect
(
)
∗∗
.
Equivalently,
in
the
language
of
Definition
1.3.12,
V
V
∗∗
naturally
in
V.
This
is
one
of
those
occasions
on
which
category
theory
makes
an
intuition
precise.
In
some
informal
sense,
evident
before
you
learn
anything
about
cat-
egory
theory,
the
isomorphism
between
a
finite-dimensional
vector
space
and
its
double
dual
is
‘natural’
or
‘canonical’:
no
arbitrary
choices
are
needed
in
order
to
define
it.
In
contrast,
to
specify
an
isomorphism
between
V
and
its
sin-
gle
dual
V
∗
,
we
need
to
make
an
arbitrary
choice
of
basis,
and
the
isomorphism
really
does
depend
on
the
basis
that
we
choose.
In
the
example
on
vector
spaces,
the
word
canonical
was
used.
It
is
an
in-
formal
word,
meaning
something
like
‘God-given’
or
‘defined
without
making
arbitrary
choices’.
For
example,
for
any
two
sets
A
and
B,
there
is
a
canonical
bijection
A
×
B
→
B
×
A
defined
by
(a,
b)
7→
(b,
a),
and
there
is
a
canonical
function
A
×
B
→
A
defined
by
(a,
b)
7→
a.
But
the
function
B
→
A
defined
by
‘choose
an
element
a
0
∈
A
and
send
everything
to
a
0
’
is
not
canonical,
because
the
choice
of
a
0
is
arbitrary.
The
concept
of
natural
isomorphism
leads
unavoidably
to
another
central
concept:
equivalence
of
categories.
Two
elements
of
a
set
are
either
equal
or
not.
Two
objects
of
a
category
can
be
equal,
not
equal
but
isomorphic,
or
not
even
isomorphic.
As
explained
be-
fore
Definition
1.3.10,
the
notion
of
equality
between
two
objects
of
a
category
is
unreasonably
strict;
it
is
usually
isomorphism
that
we
care
about.
So:
•
the
right
notion
of
sameness
of
two
elements
of
a
set
is
equality;
•
the
right
notion
of
sameness
of
two
objects
of
a
category
is
isomorphism.
When
applied
to
a
functor
category
[A
,
B],
the
second
point
tells
us
that:
•
the
right
notion
of
sameness
of
two
functors
A
⇒
B
is
natural
isomor-
phism.
But
what
is
the
right
notion
of
sameness
of
two
categories?
Isomorphism
is
unreasonably
strict,
as
if
A
B
then
there
are
functors
A
o
F
G
/
B
(1.4)
such
that
G
◦
F
=
1
A
and
F
◦
G
=
1
B
,
(1.5)
and
we
have
just
seen
that
the
notion
of
equality
between
functors
is
too
strict.
The
most
useful
notion
of
sameness
of
categories,
called
‘equivalence’,
is
34
Categories,
functors
and
natural
transformations
looser
than
isomorphism.
To
obtain
the
definition,
we
simply
replace
the
un-
reasonably
strict
equalities
in
(1.5)
by
isomorphisms.
This
gives
G
◦
F
1
A
and
F
◦
G
1
B
.
Definition
1.3.15
An
equivalence
between
categories
A
and
B
consists
of
a
pair
(1.4)
of
functors
together
with
natural
isomorphisms
η
:
1
A
→
G
◦
F,
ε
:
F
◦
G
→
1
B
.
If
there
exists
an
equivalence
between
A
and
B,
we
say
that
A
and
B
are
equivalent,
and
write
A
'
B.
We
also
say
that
the
functors
F
and
G
are
equivalences.
The
directions
of
η
and
ε
are
not
very
important,
since
they
are
isomor-
phisms
anyway.
The
reason
for
this
particular
choice
will
become
apparent
when
we
come
to
discuss
adjunctions
(Section
2.2).
Warning
1.3.16
The
symbol
is
used
for
isomorphism
of
objects
of
a
cat-
egory,
and
in
particular
for
isomorphism
of
categories
(which
are
objects
of
CAT).
The
symbol
'
is
used
for
equivalence
of
categories.
At
least,
this
is
the
convention
used
in
this
book
and
by
most
category
theorists,
although
it
is
far
from
universal
in
mathematics
at
large.
There
is
a
very
useful
alternative
characterization
of
those
functors
that
are
equivalences.
First,
we
need
a
definition.
Definition
1.3.17
A
functor
F
:
A
→
B
is
essentially
surjective
on
objects
if
for
all
B
∈
B,
there
exists
A
∈
A
such
that
F(A)
B.
Proposition
1.3.18
A
functor
is
an
equivalence
if
and
only
if
it
is
full,
faithful
and
essentially
surjective
on
objects.
Proof
Exercise
1.3.32.
This
result
can
be
compared
to
the
theorem
that
every
bijective
group
ho-
momorphism
is
an
isomorphism
(that
is,
its
inverse
is
also
a
homomorphism),
or
that
a
natural
transformation
whose
components
are
isomorphisms
is
itself
an
isomorphism
(Lemma
1.3.11).
Those
two
results
are
useful
because
they
allow
us
to
show
that
a
map
is
an
isomorphism
without
directly
constructing
an
inverse.
Proposition
1.3.18
provides
a
similar
service,
enabling
us
to
prove
that
a
functor
F
is
an
equivalence
without
actually
constructing
an
‘inverse’
G,
or
indeed
an
η
or
an
ε
(in
the
notation
of
Definition
1.3.15).
A
corollary
of
Proposition
1.3.18
invites
us
to
view
full
and
faithful
functors
as,
essentially,
inclusions
of
full
subcategories:
1.3
Natural
transformations
35
Corollary
1.3.19
Let
F
:
C
→
D
be
a
full
and
faithful
functor.
Then
C
is
equivalent
to
the
full
subcategory
C
0
of
D
whose
objects
are
those
of
the
form
F(C)
for
some
C
∈
C
.
Proof
The
functor
F
0
:
C
→
C
0
defined
by
F
0
(C)
=
F(C)
is
full
and
faithful
(since
F
is)
and
essentially
surjective
on
objects
(by
definition
of
C
0
).
This
result
is
true,
with
the
same
proof,
whether
we
interpret
‘of
the
form
F(C)’
to
mean
‘equal
to
F(C)’
or
‘isomorphic
to
F(C)’.
Example
1.3.20
Let
A
be
any
category,
and
let
B
be
any
full
subcategory
containing
at
least
one
object
from
each
isomorphism
class
of
A
.
Then
the
inclusion
functor
B
,→
A
is
faithful
(like
any
inclusion
of
subcategories),
full,
and
essentially
surjective
on
objects.
Hence
B
'
A
.
So
if
we
take
a
category
and
remove
some
(but
not
all)
of
the
objects
in
each
isomorphism
class,
the
slimmed-down
version
is
equivalent
to
the
original.
Conversely,
if
we
take
a
category
and
throw
in
some
more
objects,
each
of
them
isomorphic
to
one
of
the
existing
objects,
it
makes
no
difference:
the
new,
bigger,
category
is
equivalent
to
the
old
one.
For
example,
let
FinSet
be
the
category
of
finite
sets
and
functions
between
them.
For
each
natural
number
n,
choose
a
set
n
with
n
elements,
and
let
B
be
the
full
subcategory
of
FinSet
with
objects
0,
1,
.
.
.
.
Then
B
'
FinSet,
even
though
B
is
in
some
sense
much
smaller
than
FinSet.
Example
1.3.21
In
Example
1.1.8(d),
we
saw
that
monoids
are
essentially
the
same
thing
as
one-object
categories.
With
the
definition
of
equivalence
in
hand,
we
are
nearly
ready
to
make
this
statement
precise.
We
are
missing
some
set-theoretic
language,
and
we
will
return
to
this
result
once
we
have
that
language
(Example
3.2.11),
but
the
essential
point
can
be
stated
now.
Let
C
be
the
full
subcategory
of
CAT
whose
objects
are
the
one-object
categories.
Let
Mon
be
the
category
of
monoids.
Then
C
'
Mon.
To
see
this,
first
note
that
given
any
object
A
of
any
category,
the
maps
A
→
A
form
a
monoid
under
composition
(at
least,
subject
to
some
set-theoretic
restrictions).
There
is,
therefore,
a
canonical
functor
F
:
C
→
Mon
sending
a
one-object
category
to
the
monoid
of
maps
from
the
single
object
to
itself.
This
functor
F
is
full
and
faithful
(by
Example
1.2.7)
and
essentially
surjective
on
objects.
Hence
F
is
an
equivalence.
Example
1.3.22
An
equivalence
of
the
form
A
op
'
B
is
sometimes
called
a
duality
between
A
and
B.
One
says
that
A
is
dual
to
B.
There
are
many
famous
dualities
in
which
A
is
a
category
of
algebras
and
B
is
a
category
of
spaces;
recall
the
slogan
‘algebra
is
dual
to
geometry’
from
Example
1.2.11.
36
Categories,
functors
and
natural
transformations
Here
are
some
quite
advanced
examples,
well
beyond
the
scope
of
this
book.
•
Stone
duality:
the
category
of
Boolean
algebras
is
dual
to
the
category
of
totally
disconnected
compact
Hausdorff
spaces.
•
Gelfand–Naimark
duality:
the
category
of
commutative
unital
C
∗
-algebras
is
dual
to
the
category
of
compact
Hausdorff
spaces.
(C
∗
-algebras
are
certain
algebraic
structures
important
in
functional
analysis.)
•
Algebraic
geometers
have
several
notions
of
‘space’,
one
of
which
is
‘affine
variety’.
Let
k
be
an
algebraically
closed
field.
Then
the
category
of
affine
varieties
over
k
is
dual
to
the
category
of
finitely
generated
k-algebras
with
no
nontrivial
nilpotents.
•
Pontryagin
duality:
the
category
of
locally
compact
abelian
topological
groups
is
dual
to
itself.
As
the
words
‘topological
group’
suggest,
both
sides
of
the
duality
are
algebraic
and
geometric.
Pontryagin
duality
is
an
abstrac-
tion
of
the
properties
of
the
Fourier
transform.
Example
1.3.23
It
is
rarely
useful
to
consider
a
category
of
structured
ob-
jects
in
which
the
maps
do
not
respect
that
structure.
For
instance,
let
A
be
the
category
whose
objects
are
groups
and
whose
maps
are
all
functions
between
them,
not
necessarily
homomorphisms.
Let
Set
,∅
be
the
category
of
nonempty
sets.
The
forgetful
functor
U
:
A
→
Set
,∅
is
full
and
faithful.
It
is
a
(not
pro-
found)
fact
that
every
nonempty
set
can
be
given
at
least
one
group
structure,
so
U
is
essentially
surjective
on
objects.
Hence
U
is
an
equivalence.
This
im-
plies
that
the
category
A
,
although
defined
in
terms
of
groups,
is
really
just
the
category
of
nonempty
sets.
Remarks
1.3.24
fined:
•
•
•
•
Here
is
a
kind
of
review
of
the
chapter
so
far.
We
have
de-
categories
(Section
1.1);
functors
between
categories
(Section
1.2);
natural
transformations
between
functors
(Section
1.3);
composition
of
functors
·→·→·
and
the
identity
functor
on
any
category
(Remark
1.2.2(b));
•
composition
of
natural
transformations
·
/
·
F
and
the
identity
natural
transformation
on
any
functor
(Construction
1.3.6).
37
1.3
Natural
transformations
This
composition
of
natural
transformations
is
sometimes
called
vertical
com-
position.
There
is
also
horizontal
composition,
which
takes
natural
transfor-
mations
F
A
α
'
F
0
8
A
(
00
0
α
8
A
0
G
0
G
and
produces
a
natural
transformation
F
0
◦F
A
(
00
6
A
,
0
G
◦G
traditionally
written
as
α
0
∗
α.
The
component
of
α
0
∗
α
at
A
∈
A
is
defined
to
be
the
diagonal
of
the
naturality
square
F
0
(F(A))
F
0
(α
A
)
α
0
F(A)
G
0
(F(A))
/
F
0
(G(A))
0
α
G(A)
G
0
(α
A
)
/
G
0
(G(A)).
0
In
other
words,
(α
0
∗α)
A
can
be
defined
as
either
α
G(A)
◦F
0
(α
A
)
or
G
0
(α
A
)◦α
0
F(A)
;
it
makes
no
difference
which,
since
they
are
equal.
The
special
cases
of
horizontal
composition
where
either
α
or
α
0
is
an
iden-
tity
are
especially
important,
and
have
their
own
notation.
Thus,
F
0
A
/
A
0
F
α
0
)
F
0
◦F
6
A
00
gives
rise
to
A
G
0
(
α
0
F
6
A
00
6
A
00
G
0
◦F
where
(α
0
F)
A
=
α
0
F(A)
,
and
F
A
α
(
F
0
◦F
6
A
0
F
0
G
where
(F
0
α)
A
=
F
0
(α
A
).
/
A
00
gives
rise
to
A
F
0
α
F
0
◦G
(
38
Categories,
functors
and
natural
transformations
Vertical
and
horizontal
composition
interact
well:
natural
transformations
F
0
F
A
α
β
G
H
/
A
0
E
/
A
00
E
0
α
0
G
0
β
H
0
obey
the
interchange
law,
(β
0
◦
α
0
)
∗
(β
◦
α)
=
(β
0
∗
β)
◦
(α
0
∗
α)
:
F
0
◦
F
→
H
0
◦
H.
As
usual,
a
statement
on
composition
is
accompanied
by
a
statement
on
iden-
tities:
1
F
0
∗
1
F
=
1
F
0
◦F
too.
All
of
this
enables
us
to
construct,
for
any
categories
A
,
A
0
and
A
00
,
a
functor
[A
0
,
A
00
]
×
[A
,
A
0
]
→
[A
,
A
00
],
given
on
objects
by
(F
0
,
F)
7→
F
0
◦
F
and
on
maps
by
(α
0
,
α)
7→
α
0
∗
α.
In
particular,
if
F
0
G
0
and
F
G
then
F
0
◦
F
G
0
◦
G,
since
functors
preserve
isomorphism
(Exercise
1.2.21).
(The
existence
of
this
functor
is
similar
to
the
fact
that
inside
a
category
C
,
we
have,
for
any
objects
A,
A
0
and
A
00
,
a
function
C
(A
0
,
A
00
)
×
C
(A,
A
0
)
→
C
(A,
A
00
),
given
by
(
f
0
,
f
)
7→
f
0
◦
f
.)
The
diagrams
above
contain
not
only
objects
(0-dimensional)
and
arrows
→
(1-dimensional),
but
also
double
arrows
⇒
sweeping
out
2-dimensional
re-
gions
between
arrows.
What
we
are
implicitly
doing
is
called
2-category
the-
ory.
There
is
a
2-category
of
categories,
functors
and
natural
transformations,
whose
anatomy
we
have
just
been
describing.
If
we
are
really
serious
about
categories,
we
have
to
get
serious
about
2-categories.
And
if
we
are
really
seri-
ous
about
2-categories,
we
have
to
get
serious
about
3-categories.
.
.
and
before
we
know
it,
we
are
studying
∞-categories.
But
in
this
book,
we
climb
no
higher
than
the
first
rung
or
two
of
this
infinite
ladder.
Exercises
1.3.25
Find
three
examples
of
natural
transformations
not
mentioned
above.
1.3.26
Prove
Lemma
1.3.11.
1.3
Natural
transformations
39
1.3.27
Let
A
and
B
be
categories.
Prove
that
[A
op
,
B
op
]
[A
,
B]
op
.
1.3.28
Let
A
and
B
be
sets,
and
denote
by
B
A
the
set
of
functions
from
A
to
B.
Write
down:
(a)
a
canonical
function
A
×
B
A
→
B;
A
(b)
a
canonical
function
A
→
B
(B
)
.
(Although
in
principle
there
could
be
many
such
canonical
functions,
in
both
these
cases
there
is
only
one.)
1.3.29
Here
we
consider
natural
transformations
between
functors
whose
do-
main
is
a
product
category
A
×
B.
Your
task
is
to
show
that
naturality
in
two
variables
simultaneously
is
equivalent
to
naturality
in
each
variable
separately.
Take
functors
F,
G
:
A
×
B
→
C
.
For
each
A
∈
A
,
there
are
functors
F
A
,
G
A
:
B
→
C
,
as
in
Exercise
1.2.25.
Similarly,
for
each
B
∈
B,
there
are
functors
F
B
,
G
B
:
A
→
C
.
Let
α
A,B
:
F(A,
B)
→
G(A,
B)
A∈A
,B∈B
be
a
family
of
maps.
Show
that
this
family
is
a
natural
transformation
F
→
G
if
and
only
if
it
satisfies
the
following
two
conditions:
•
for
each
A
∈
A
,
the
family
α
A,B
:
F
A
(B)
→
G
A
(B)
B∈B
is
a
natural
trans-
formation
F
A
→
G
A
;
•
for
each
B
∈
B,
the
family
α
A,B
:
F
B
(A)
→
G
B
(A)
A∈A
is
a
natural
trans-
formation
F
B
→
G
B
.
1.3.30
Let
G
be
a
group.
For
each
g
∈
G,
there
is
a
unique
homomorphism
φ
:
Z
→
G
satisfying
φ(1)
=
g.
Thus,
elements
of
G
are
essentially
the
same
thing
as
homomorphisms
Z
→
G.
When
groups
are
regarded
as
one-object
categories,
homomorphisms
Z
→
G
are
in
turn
the
same
as
functors
Z
→
G.
Natural
isomorphism
defines
an
equivalence
relation
on
the
set
of
functors
Z
→
G,
and,
therefore,
an
equivalence
relation
on
G
itself.
What
is
this
equivalence
relation,
in
purely
group-theoretic
terms?
(First
have
a
guess.
For
a
general
group
G,
what
equivalence
relations
on
G
can
you
think
of?)
1.3.31
A
permutation
of
a
set
X
is
a
bijection
X
→
X.
Write
Sym(X)
for
the
set
of
permutations
of
X.
A
total
order
on
a
set
X
is
an
order
≤
such
that
for
all
x,
y
∈
X,
either
x
≤
y
or
y
≤
x;
so
a
total
order
on
a
finite
set
amounts
to
a
way
of
placing
its
elements
in
sequence.
Write
Ord(X)
for
the
set
of
total
orders
on
X.
Let
B
denote
the
category
of
finite
sets
and
bijections.
40
Categories,
functors
and
natural
transformations
(a)
Give
a
definition
of
Sym
on
maps
in
B
in
such
a
way
that
Sym
becomes
a
functor
B
→
Set.
Do
the
same
for
Ord.
Both
your
definitions
should
be
canonical
(no
arbitrary
choices).
(b)
Show
that
there
is
no
natural
transformation
Sym
→
Ord.
(Hint:
consider
identity
permutations.)
(c)
For
an
n-element
set
X,
how
many
elements
do
the
sets
Sym(X)
and
Ord(X)
have?
Conclude
that
Sym(X)
Ord(X)
for
all
X
∈
B,
but
not
naturally
in
X
∈
B.
(The
moral
is
that
for
each
finite
set
X,
there
are
exactly
as
many
permutations
of
X
as
there
are
total
orders
on
X,
but
there
is
no
natural
way
of
matching
them
up.)
1.3.32
In
this
exercise,
you
will
prove
Proposition
1.3.18.
Let
F
:
A
→
B
be
a
functor.
(a)
Suppose
that
F
is
an
equivalence.
Prove
that
F
is
full,
faithful
and
essen-
tially
surjective
on
objects.
(Hint:
prove
faithfulness
before
fullness.)
(b)
Now
suppose
instead
that
F
is
full,
faithful
and
essentially
surjective
on
objects.
For
each
B
∈
B,
choose
an
object
G(B)
of
A
and
an
isomorphism
ε
B
:
F(G(B))
→
B.
Prove
that
G
extends
to
a
functor
in
such
a
way
that
(ε
B
)
B∈B
is
a
natural
isomorphism
FG
→
1
B
.
Then
construct
a
natural
isomorphism
1
A
→
GF,
thus
proving
that
F
is
an
equivalence.
1.3.33
This
exercise
makes
precise
the
idea
that
linear
algebra
can
equiva-
lently
be
done
with
matrices
or
with
linear
maps.
Fix
a
field
k.
Let
Mat
be
the
category
whose
objects
are
the
natural
numbers
and
with
Mat(m,
n)
=
{n
×
m
matrices
over
k}.
Prove
that
Mat
is
equivalent
to
FDVect,
the
category
of
finite-dimensional
vector
spaces
over
k.
Does
your
equivalence
involve
a
canonical
functor
from
Mat
to
FDVect,
or
from
FDVect
to
Mat?
(Part
of
the
exercise
is
to
work
out
what
composition
in
the
category
Mat
is
supposed
to
be;
there
is
only
one
sensible
possibility.
Proposition
1.3.18
makes
the
exercise
easier.)
1.3.34
Show
that
equivalence
of
categories
is
an
equivalence
relation.
(Not
as
obvious
as
it
looks.)
2
Adjoints
The
slogan
of
Saunders
Mac
Lane’s
book
Categories
for
the
Working
Mathe-
matician
is:
Adjoint
functors
arise
everywhere.
We
will
see
the
truth
of
this,
meeting
examples
of
adjoint
functors
from
diverse
parts
of
mathematics.
To
complement
the
understanding
provided
by
exam-
ples,
we
will
approach
the
theory
of
adjoints
from
three
different
directions,
each
of
which
carries
its
own
intuition.
Then
we
will
prove
that
the
three
ap-
proaches
are
equivalent.
Understanding
adjointness
gives
you
a
valuable
addition
to
your
mathemat-
ical
toolkit.
Most
professional
pure
mathematicians
know
what
categories
and
functors
are,
but
far
fewer
know
about
adjoints.
More
should:
adjoint
func-
tors
are
both
common
and
easy,
and
knowing
about
adjoints
helps
you
to
spot
patterns
in
the
mathematical
landscape.
2.1
Definition
and
examples
Consider
a
pair
of
functors
in
opposite
directions,
F
:
A
→
B
and
G
:
B
→
A
.
Roughly
speaking,
F
is
said
to
be
left
adjoint
to
G
if,
whenever
A
∈
A
and
B
∈
B,
maps
F(A)
→
B
are
essentially
the
same
thing
as
maps
A
→
G(B).
Definition
2.1.1
Let
A
o
F
G
/
B
be
categories
and
functors.
We
say
that
F
is
left
adjoint
to
G,
and
G
is
right
adjoint
to
F,
and
write
F
a
G,
if
B(F(A),
B)
A
(A,
G(B))
(2.1)
naturally
in
A
∈
A
and
B
∈
B.
The
meaning
of
‘naturally’
is
defined
below.
An
adjunction
between
F
and
G
is
a
choice
of
natural
isomorphism
(2.1).
41
42
Adjoints
‘Naturally
in
A
∈
A
and
B
∈
B’
means
that
there
is
a
specified
bijec-
tion
(2.1)
for
each
A
∈
A
and
B
∈
B,
and
that
it
satisfies
a
naturality
axiom.
To
state
it,
we
need
some
notation.
Given
objects
A
∈
A
and
B
∈
B,
the
cor-
respondence
(2.1)
between
maps
F(A)
→
B
and
A
→
G(B)
is
denoted
by
a
horizontal
bar,
in
both
directions:
ḡ
g
F(A)
−→
B
7→
A
−→
G(B)
,
f
f
¯
F(A)
−→
B
→
7
A
−→
G(B)
.
So
f
¯
=
f
and
ḡ
¯
=
g.
We
call
f
¯
the
transpose
of
f
,
and
similarly
for
g.
The
naturality
axiom
has
two
parts:
ḡ
q
G(q)
g
F(A)
−→
B
−→
B
0
=
A
−→
G(B)
−→
G(B
0
)
(2.2)
(that
is,
q
◦
g
=
G(q)
◦
ḡ)
for
all
g
and
q,
and
p
f
A
0
−→
A
−→
G(B)
=
F(p)
f
¯
F(A
0
)
−→
F(A)
−→
B
(2.3)
for
all
p
and
f
.
It
makes
no
difference
whether
we
put
the
long
bar
over
the
left
or
the
right
of
these
equations,
since
bar
is
self-inverse.
Remarks
2.1.2
(a)
The
naturality
axiom
might
seem
ad
hoc,
but
we
will
see
in
Chapter
4
that
it
simply
says
that
two
particular
functors
are
natu-
rally
isomorphic.
In
this
section,
we
ignore
the
naturality
axiom
altogether,
trusting
that
it
embodies
our
usual
intuitive
idea
of
naturality:
something
defined
without
making
any
arbitrary
choices.
(b)
The
naturality
axiom
implies
that
from
each
array
of
maps
A
0
→
·
·
·
→
A
n
,
F(A
n
)
→
B
0
,
B
0
→
·
·
·
→
B
m
,
it
is
possible
to
construct
exactly
one
map
A
0
→
G(B
m
).
Compare
the
comments
on
the
definitions
of
category,
functor
and
natural
transformation
(Remarks
1.1.2(b),
1.2.2(a),
and
1.3.2(a)).
(c)
Not
only
do
adjoint
functors
arise
everywhere;
better,
whenever
you
see
a
pair
of
functors
A
B,
there
is
an
excellent
chance
that
they
are
adjoint
(one
way
round
or
the
other).
For
example,
suppose
you
get
talking
to
a
mathematician
who
tells
you
that
her
work
involves
Lie
algebras
and
associative
algebras.
You
try
to
object
that
you
don’t
know
what
either
of
those
things
is,
but
she
carries
on
talking
anyway,
explaining
that
there’s
a
way
of
turning
any
Lie
alge-
bra
into
an
associative
algebra,
and
also
a
way
of
turning
any
associative
2.1
Definition
and
examples
43
algebra
into
a
Lie
algebra.
At
this
point,
even
without
knowing
what
she’s
talking
about,
you
should
bet
her
that
one
process
is
adjoint
to
the
other.
This
almost
always
works.
(d)
A
given
functor
G
may
or
may
not
have
a
left
adjoint,
but
if
it
does,
it
is
unique
up
to
isomorphism,
so
we
may
speak
of
‘the
left
adjoint
of
G’.
The
same
goes
for
right
adjoints.
We
prove
this
later
(Example
4.3.13).
You
might
ask
‘what
do
we
gain
from
knowing
that
two
functors
are
adjoint?’
The
uniqueness
is
a
crucial
part
of
the
answer.
Let
us
return
to
the
example
of
(c).
It
would
take
you
only
a
few
minutes
to
learn
what
Lie
algebras
are,
what
associative
algebras
are,
and
what
the
standard
functor
G
is
that
turns
an
associative
algebra
into
a
Lie
algebra.
What
about
the
functor
F
in
the
opposite
direction?
The
description
of
F
that
you
will
find
in
most
algebra
books
(under
‘universal
enveloping
algebra’)
takes
much
longer
to
understand.
However,
you
can
bypass
that
process
completely,
just
by
knowing
that
F
is
the
left
adjoint
of
G.
Since
G
can
have
only
one
left
adjoint,
this
characterizes
F
completely.
In
a
sense,
it
tells
you
all
you
need
to
know.
Examples
2.1.3
(Algebra:
free
a
forgetful)
Forgetful
functors
between
cat-
egories
of
algebraic
structures
usually
have
left
adjoints.
For
instance:
(a)
Let
k
be
a
field.
There
is
an
adjunction
Vect
O
k
F
a
U
Set,
where
U
is
the
forgetful
functor
of
Example
1.2.3(b)
and
F
is
the
free
functor
of
Example
1.2.4(c).
Adjointness
says
that
given
a
set
S
and
a
vector
space
V,
a
linear
map
F(S
)
→
V
is
essentially
the
same
thing
as
a
function
S
→
U(V).
We
saw
this
in
Example
0.4,
but
let
us
now
check
it
in
detail.
Fix
a
set
S
and
a
vector
space
V.
Given
a
linear
map
g
:
F(S
)
→
V,
we
may
define
a
map
of
sets
ḡ
:
S
→
U(V)
by
ḡ(s)
=
g(s)
for
all
s
∈
S
.
This
gives
a
function
Vect
k
(F(S
),
V)
→
g
7→
Set(S
,
U(V))
ḡ.
In
the
other
direction,
given
a
map
of
sets
f
:
S
→
U(V),
we
may
define
P
P
a
linear
map
f
¯
:
F(S
)
→
V
by
f
¯
s∈S
λ
s
s
=
s∈S
λ
s
f
(s)
for
all
formal
44
Adjoints
linear
combinations
P
λ
s
s
∈
F(S
).
This
gives
a
function
Set(S
,
U(V))
→
f
7→
Vect
k
(F(S
),
V)
f
¯
.
These
two
functions
‘bar’
are
mutually
inverse:
for
any
linear
map
g
:
F(S
)
→
V,
we
have
X
!
X
X
X
!
ḡ
¯
λ
s
s
=
λ
s
ḡ(s)
=
λ
s
g(s)
=
g
λ
s
s
s∈S
for
all
have
P
s∈S
s∈S
s∈S
λ
s
s
∈
F(S
),
so
ḡ
¯
=
g,
and
for
any
map
of
sets
f
:
S
→
U(V),
we
f
¯
(s)
=
f
¯
(s)
=
f
(s)
for
all
s
∈
S
,
so
f
¯
=
f
.
We
therefore
have
a
canonical
bijection
between
Vect
k
(F(S
),
V)
and
Set(S
,
U(V))
for
each
S
∈
Set
and
V
∈
Vect
k
,
as
re-
quired.
Here
we
have
been
careful
to
distinguish
between
the
vector
space
V
and
its
underlying
set
U(V).
Very
often,
though,
in
category
theory
as
in
mathematics
at
large,
the
symbol
for
a
forgetful
functor
is
omitted.
In
this
example,
that
would
mean
dropping
the
U
and
leaving
the
reader
to
figure
out
whether
each
occurrence
of
V
is
intended
to
denote
the
vector
space
it-
self
or
its
underlying
set.
We
will
soon
start
using
such
notational
shortcuts
ourselves.
(b)
In
the
same
way,
there
is
an
adjunction
Grp
O
F
a
U
Set
where
F
and
U
are
the
free
and
forgetful
functors
of
Examples
1.2.3(a)
and
1.2.4(a).
The
free
group
functor
is
tricky
to
construct
explicitly.
In
Chapter
6,
we
will
prove
a
result
(the
general
adjoint
functor
theorem)
guaranteeing
that
U
and
many
functors
like
it
all
have
left
adjoints.
To
some
extent,
this
removes
the
need
to
construct
F
explicitly,
as
observed
in
Remark
2.1.2(d).
The
point
can
be
overstated:
for
a
group
theorist,
the
more
descriptions
of
free
groups
that
are
available,
the
better.
Explicit
constructions
really
can
be
useful.
But
it
is
an
important
general
principle
that
forgetful
functors
of
this
type
always
have
left
adjoints.
2.1
Definition
and
examples
45
(c)
There
is
an
adjunction
Ab
O
F
a
U
Grp
where
U
is
the
inclusion
functor
of
Example
1.2.3(d).
If
G
is
a
group
then
F(G)
is
the
abelianization
G
ab
of
G.
This
is
an
abelian
quotient
group
of
G,
with
the
property
that
every
map
from
G
to
an
abelian
group
factorizes
uniquely
through
G
ab
:
η
G
∀φ
/
G
ab
∃!
φ̄
∀A.
Here
η
is
the
natural
map
from
G
to
its
quotient
G
ab
,
and
A
is
any
abelian
group.
(We
have
adopted
the
abuse
of
notation
advertised
in
example
(a),
omitting
the
symbol
U
at
several
places
in
this
diagram.)
The
bijection
Ab(G
ab
,
A)
Grp(G,
U(A))
is
given
in
the
left-to-right
direction
by
ψ
7→
ψ
◦
η,
and
in
the
right-to-left
direction
by
φ
7→
φ̄.
(To
construct
G
ab
,
let
G
0
be
the
smallest
normal
subgroup
of
G
contain-
ing
xyx
−1
y
−1
for
all
x,
y
∈
G,
and
put
G
ab
=
G/G
0
.
The
kernel
of
any
homomorphism
from
G
to
an
abelian
group
contains
G
0
,
and
the
universal
property
follows.)
(d)
There
are
adjunctions
O
Grp
O
F
a
U
a
R
Mon
between
the
categories
of
groups
and
monoids.
The
middle
functor
U
is
inclusion.
The
left
adjoint
F
is,
again,
tricky
to
describe
explicitly.
Infor-
mally,
F(M)
is
obtained
from
M
by
throwing
in
an
inverse
to
every
ele-
ment.
(For
example,
if
M
is
the
additive
monoid
of
natural
numbers
then
F(M)
is
the
group
of
integers.)
Again,
the
general
adjoint
functor
theorem
(Theorem
6.3.10)
guarantees
the
existence
of
this
adjoint.
This
example
is
unusual
in
that
forgetful
functors
do
not
usually
have
right
adjoints.
Here,
given
a
monoid
M,
the
group
R(M)
is
the
submonoid
of
M
consisting
of
all
the
invertible
elements.
46
Adjoints
The
category
Grp
is
both
a
reflective
and
a
coreflective
subcategory
of
Mon.
This
means,
by
definition,
that
the
inclusion
functor
Grp
,→
Mon
has
both
a
left
and
a
right
adjoint.
The
previous
example
tells
us
that
Ab
is
a
reflective
subcategory
of
Grp.
(e)
Let
Field
be
the
category
of
fields,
with
ring
homomorphisms
as
the
maps.
The
forgetful
functor
Field
→
Set
does
not
have
a
left
adjoint.
(For
a
proof,
see
Example
6.3.5.)
The
theory
of
fields
is
unlike
the
theories
of
groups,
rings,
and
so
on,
because
the
operation
x
7→
x
−1
is
not
defined
for
all
x
(only
for
x
,
0).
Remark
2.1.4
At
several
points
in
this
book,
we
make
contact
with
the
idea
of
an
algebraic
theory.
You
already
know
several
examples:
the
theory
of
groups
is
an
algebraic
theory,
as
are
the
theory
of
rings,
the
theory
of
vector
spaces
over
R,
the
theory
of
vector
spaces
over
C,
the
theory
of
monoids,
and
(rather
trivially)
the
theory
of
sets.
After
reading
the
description
below,
you
might
conclude
that
the
word
‘theory’
is
overly
grand,
and
that
‘definition’
would
be
more
appropriate.
Nevertheless,
this
is
the
established
usage.
We
will
not
need
to
define
‘algebraic
theory’
formally,
but
it
will
be
impor-
tant
to
have
the
general
idea.
Let
us
begin
by
considering
the
theory
of
groups.
A
group
can
be
defined
as
a
set
X
equipped
with
a
function
·
:
X
×
X
→
X
(multiplication),
another
function
(
)
−1
:
X
→
X
(inverse),
and
an
element
e
∈
X
(the
identity),
satisfying
a
familiar
list
of
equations.
More
systematically,
the
three
pieces
of
structure
on
X
can
be
seen
as
maps
of
sets
·
:
X
2
→
X,
(
)
−1
:
X
1
→
X,
e
:
X
0
→
X,
where
in
the
last
case,
X
0
is
the
one-element
set
1
and
we
are
using
the
obser-
vation
that
a
map
1
→
X
of
sets
is
essentially
the
same
thing
as
an
element
of
X.
(You
may
be
more
familiar
with
a
definition
of
group
in
which
only
the
multiplication
and
perhaps
the
identity
are
specified
as
pieces
of
structure,
with
the
existence
of
inverses
required
as
a
property.
In
that
approach,
the
definition
is
swiftly
followed
by
a
lemma
on
uniqueness
of
inverses,
guaranteeing
that
it
makes
sense
to
speak
of
the
inverse
of
an
element.
The
two
approaches
are
equivalent,
but
for
many
purposes,
it
is
better
to
frame
the
definition
in
the
way
described
in
the
previous
paragraph.)
An
algebraic
theory
consists
of
two
things:
first,
a
collection
of
operations,
each
with
a
specified
arity
(number
of
inputs),
and
second,
a
collection
of
equa-
tions.
For
example,
the
theory
of
groups
has
one
operation
of
arity
2,
one
of
arity
1,
and
one
of
arity
0.
An
algebra
or
model
for
an
algebraic
theory
con-
sists
of
a
set
X
together
with
a
specified
map
X
n
→
X
for
each
operation
of
2.1
Definition
and
examples
47
arity
n,
such
that
the
equations
hold
everywhere.
For
example,
an
algebra
for
the
theory
of
groups
is
exactly
a
group.
A
more
subtle
example
is
the
theory
of
vector
spaces
over
R.
This
is
an
algebraic
theory
with,
among
other
things,
an
infinite
number
of
operations
of
arity
1:
for
each
λ
∈
R,
we
have
the
operation
λ
·
−
:
X
→
X
of
scalar
multiplication
by
λ
(for
any
vector
space
X).
There
is
nothing
special
about
the
field
R
here;
the
only
point
is
that
it
was
chosen
in
advance.
The
theory
of
vector
spaces
over
R
is
different
from
the
theory
of
vector
spaces
over
C,
because
they
have
different
operations
of
arity
1.
In
a
nutshell,
the
main
property
of
algebras
for
an
algebraic
theory
is
that
the
operations
are
defined
everywhere
on
the
set,
and
the
equations
hold
every-
where
too.
For
example,
every
element
of
a
group
has
a
specified
inverse,
and
every
element
x
satisfies
the
equation
x
·
x
−1
=
1.
This
is
why
the
theories
of
groups,
rings,
and
so
on,
are
algebraic
theories,
but
the
theory
of
fields
is
not.
Example
2.1.5
There
are
adjunctions
O
Top
O
D
a
U
a
I
Set
where
U
sends
a
space
to
its
set
of
points,
D
equips
a
set
with
the
discrete
topology,
and
I
equips
a
set
with
the
indiscrete
topology.
Example
2.1.6
Given
sets
A
and
B,
we
can
form
their
(cartesian)
product
A
×
B.
We
can
also
form
the
set
B
A
of
functions
from
A
to
B.
This
is
the
same
as
the
set
Set(A,
B),
but
we
tend
to
use
the
notation
B
A
when
we
want
to
emphasize
that
it
is
an
object
of
the
same
category
as
A
and
B.
Now
fix
a
set
B.
Taking
the
product
with
B
defines
a
functor
−
×
B
:
Set
→
A
7→
Set
A
×
B.
(Here
we
are
using
the
blank
notation
introduced
in
Example
1.2.12.)
There
is
also
a
functor
(−)
B
:
Set
→
Set
C
7→
C
B
.
Moreover,
there
is
a
canonical
bijection
Set(A
×
B,
C)
Set(A,
C
B
)
for
any
sets
A
and
C.
It
is
defined
by
simply
changing
the
punctuation:
given
a
48
Adjoints
C
B
A
Figure
2.1
In
Set,
a
map
A
×
B
→
C
can
be
seen
as
a
way
of
assigning
to
each
element
of
A
a
map
B
→
C.
map
g
:
A
×
B
→
C,
define
ḡ
:
A
→
C
B
by
(ḡ(a))(b)
=
g(a,
b)
(a
∈
A,
b
∈
B),
and
in
the
other
direction,
given
f
:
A
→
C
B
,
define
f
¯
:
A
×
B
→
C
by
f
¯
(a,
b)
=
(
f
(a))(b)
(a
∈
A,
b
∈
B).
Figure
2.1
shows
an
example
with
A
=
B
=
C
=
R.
By
slicing
up
the
surface
as
shown,
a
map
R
2
→
R
can
be
seen
as
a
map
from
R
to
{maps
R
→
R}.
Putting
all
this
together,
we
obtain
an
adjunction
Set
O
−×B
a
(−)
B
Set
for
every
set
B.
Definition
2.1.7
Let
A
be
a
category.
An
object
I
∈
A
is
initial
if
for
every
A
∈
A
,
there
is
exactly
one
map
I
→
A.
An
object
T
∈
A
is
terminal
if
for
every
A
∈
A
,
there
is
exactly
one
map
A
→
T
.
For
example,
the
empty
set
is
initial
in
Set,
the
trivial
group
is
initial
in
Grp,
and
Z
is
initial
in
Ring
(Example
0.2).
The
one-element
set
is
terminal
in
Set,
the
trivial
group
is
terminal
(as
well
as
initial)
in
Grp,
and
the
trivial
(one-
element)
ring
is
terminal
in
Ring.
The
terminal
object
of
CAT
is
the
category
1
containing
just
one
object
and
one
map
(necessarily
the
identity
on
that
object).
A
category
need
not
have
an
initial
object,
but
if
it
does
have
one,
it
is
unique
up
to
isomorphism.
Indeed,
it
is
unique
up
to
unique
isomorphism,
as
follows.
2.1
Definition
and
examples
49
Lemma
2.1.8
Let
I
and
I
0
be
initial
objects
of
a
category.
Then
there
is
a
unique
isomorphism
I
→
I
0
.
In
particular,
I
I
0
.
Proof
Since
I
is
initial,
there
is
a
unique
map
f
:
I
→
I
0
.
Since
I
0
is
initial,
there
is
a
unique
map
f
0
:
I
0
→
I.
Now
f
0
◦
f
and
1
I
are
both
maps
I
→
I,
and
I
is
initial,
so
f
0
◦
f
=
1
I
.
Similarly,
f
◦
f
0
=
1
I
0
.
Hence
f
is
an
isomorphism,
as
required.
Example
2.1.9
Initial
and
terminal
objects
can
be
described
as
adjoints.
Let
A
be
a
category.
There
is
precisely
one
functor
A
→
1.
Also,
a
functor
1
→
A
is
essentially
just
an
object
of
A
(namely,
the
object
to
which
the
unique
object
of
1
is
mapped).
Viewing
functors
1
→
A
as
objects
of
A
,
a
left
adjoint
to
A
→
1
is
exactly
an
initial
object
of
A
.
Similarly,
a
right
adjoint
to
the
unique
functor
A
→
1
is
exactly
a
terminal
object
of
A
.
Remark
2.1.10
In
the
language
introduced
in
Remark
1.1.10,
the
concept
of
terminal
object
is
dual
to
the
concept
of
initial
object.
(More
generally,
the
concepts
of
left
and
right
adjoint
are
dual
to
one
another.)
Since
any
two
initial
objects
of
a
category
are
uniquely
isomorphic,
the
principle
of
duality
implies
that
the
same
is
true
of
terminal
objects.
Remark
2.1.11
Adjunctions
can
be
composed.
Take
adjunctions
A
o
F
⊥
G
/
F
0
⊥
A
0
o
G
/
A
00
0
where
the
⊥
symbol
is
a
rotated
a
(thus,
F
a
G
and
F
0
a
G
0
).
Then
we
obtain
an
adjunction
A
o
F
0
◦F
⊥
/
A
00
,
G◦G
0
since
for
A
∈
A
and
A
00
∈
A
00
,
A
00
F
0
(F(A)),
A
00
A
0
F(A),
G
0
(A
00
)
A
A,
G(G
0
(A
00
))
naturally
in
A
and
A
00
.
Exercises
2.1.12
Find
three
examples
of
adjoint
functors
not
mentioned
above.
Do
the
same
for
initial
and
terminal
objects.
2.1.13
What
can
be
said
about
adjunctions
between
discrete
categories?
50
Adjoints
2.1.14
Show
that
the
naturality
equations
(2.2)
and
(2.3)
can
equivalently
be
replaced
by
the
single
equation
G(q)
p
f
A
0
−→
A
−→
G(B)
−→
G(B
0
)
=
F(p)
f
¯
q
F(A
0
)
−→
F(A)
−→
B
−→
B
0
for
all
p,
f
and
q.
2.1.15
Show
that
left
adjoints
preserve
initial
objects:
that
is,
if
A
o
F
⊥
/
B
G
and
I
is
an
initial
object
of
A
,
then
F(I)
is
an
initial
object
of
B.
Dually,
show
that
right
adjoints
preserve
terminal
objects.
(In
Section
6.3,
we
will
see
this
as
part
of
a
bigger
picture:
right
adjoints
preserve
limits
and
left
adjoints
preserve
colimits.)
2.1.16
Let
G
be
a
group.
(a)
What
interesting
functors
are
there
(in
either
direction)
between
Set
and
the
category
[G,
Set]
of
left
G-sets?
Which
of
those
functors
are
adjoint
to
which?
(b)
Similarly,
what
interesting
functors
are
there
between
Vect
k
and
the
cate-
gory
[G,
Vect
k
]
of
k-linear
representations
of
G,
and
what
adjunctions
are
there
between
those
functors?
2.1.17
Fix
a
topological
space
X,
and
write
O(X)
for
the
poset
of
open
sub-
sets
of
X,
ordered
by
inclusion.
Let
∆
:
Set
→
[O(X)
op
,
Set]
be
the
functor
assigning
to
a
set
A
the
presheaf
∆A
with
constant
value
A.
Exhibit
a
chain
of
adjoint
functors
Λ
a
Π
a
∆
a
Γ
a
∇.
2.2
Adjunctions
via
units
and
counits
In
the
previous
section,
we
met
the
definition
of
adjunction.
In
this
section
and
the
next,
we
meet
two
ways
of
rephrasing
the
definition.
The
one
in
this
section
is
most
useful
for
theoretical
purposes,
while
the
one
in
the
next
fits
well
with
many
examples.
To
start
building
the
theory
of
adjoint
functors,
we
have
to
take
seriously
the
naturality
requirement
(equations
(2.2)
and
(2.3)),
which
has
so
far
been
2.2
Adjunctions
via
units
and
counits
ignored.
Take
an
adjunction
A
o
/
F
⊥
51
B
.
Intuitively,
naturality
says
that
as
A
G
varies
in
A
and
B
varies
in
B,
the
isomorphism
between
B(F(A),
B)
and
A
(A,
G(B))
varies
in
a
way
that
is
compatible
with
all
the
structure
already
in
place.
In
other
words,
it
is
compatible
with
composition
in
the
categories
A
and
B
and
the
action
of
the
functors
F
and
G.
But
what
does
‘compatible’
mean?
Suppose,
for
example,
that
we
have
maps
g
q
F(A)
−→
B
−→
B
0
in
B.
There
are
two
things
we
can
do
with
this
data:
either
compose
then
take
the
transpose,
which
produces
a
map
q
◦
g
:
A
→
G(B
0
),
or
take
the
transpose
of
g
then
compose
it
with
G(q),
which
produces
a
potentially
different
map
G(q)
◦
ḡ
:
A
→
G(B
0
).
Compatibility
means
that
they
are
equal;
and
that
is
the
first
naturality
equation
(2.2).
The
second
is
its
dual,
and
can
be
explained
in
a
similar
way.
For
each
A
∈
A
,
we
have
a
map
η
A
1
A
−→
GF(A)
=
F(A)
−→
F(A)
.
Dually,
for
each
B
∈
B,
we
have
a
map
ε
B
1
FG(B)
−→
B
=
G(B)
−→
G(B)
.
(We
have
begun
to
omit
brackets,
writing
GF(A)
instead
of
G(F(A)),
etc.)
These
define
natural
transformations
η
:
1
A
→
G
◦
F,
ε
:
F
◦
G
→
1
B
,
called
the
unit
and
counit
of
the
adjunction,
respectively.
Example
2.2.1
Take
the
usual
adjunction
Vect
k
o
U
>
/
Set
.
Its
unit
η
:
1
Set
→
F
U
◦
F
has
components
η
S
:
S
s
→
7→
P
UF(S
)
=
formal
k-linear
sums
s∈S
λ
s
s
s
(S
∈
Set).
The
component
of
the
counit
ε
at
a
vector
space
V
is
the
linear
map
ε
V
:
FU(V)
→
V
P
that
sends
a
formal
linear
sum
v∈V
λ
v
v
to
its
actual
value
in
V.
The
vector
space
FU(V)
is
enormous.
For
instance,
if
k
=
R
and
V
is
the
vector
space
R
2
,
then
U(V)
is
the
set
R
2
and
FU(V)
is
a
vector
space
with
52
Adjoints
one
basis
element
for
every
element
of
R
2
;
thus,
it
is
uncountably
infinite-
dimensional.
Then
ε
V
is
a
map
from
this
infinite-dimensional
space
to
the
2-
dimensional
space
V.
Lemma
2.2.2
gles
Given
an
adjunction
F
a
G
with
unit
η
and
counit
ε,
the
trian-
/
FGF
Fη
F
"
F
1
F
G
εF
ηG
/
GFG
Gε
"
G
1
G
commute.
Remark
2.2.3
These
are
called
the
triangle
identities.
They
are
commuta-
tive
diagrams
in
the
functor
categories
[A
,
B]
and
[B,
A
],
respectively.
For
an
explanation
of
the
notation,
see
Remarks
1.3.24
(particularly
the
special
cases
mentioned
on
page
37).
An
equivalent
statement
is
that
the
triangles
F(A)
F(η
A
)
/
FGF(A)
$
1
F(A)
G(B)
ε
F(A)
η
G(B)
1
G(B)
F(A)
/
GFG(B)
$
G(ε
B
)
G(B)
(2.4)
commute
for
all
A
∈
A
and
B
∈
B.
Proof
of
Lemma
2.2.2
We
prove
that
the
triangles
(2.4)
commute.
Let
A
∈
A
.
Since
1
GF(A)
=
ε
F(A)
,
equation
(2.3)
gives
η
A
1
A
−→
GF(A)
−→
GF(A)
=
ε
F(A)
F(η
A
)
F(A)
−→
FGF(A)
−→
F(A)
.
But
the
left-hand
side
is
η
A
=
1
F(A)
=
1
F(A)
,
proving
the
first
identity.
The
second
follows
by
duality.
Amazingly,
the
unit
and
counit
determine
the
whole
adjunction,
even
though
they
appear
to
know
only
the
transposes
of
identities.
This
is
the
main
content
of
the
following
pair
of
results.
Lemma
2.2.4
Let
A
o
F
⊥
G
/
B
be
an
adjunction,
with
unit
η
and
counit
ε.
Then
ḡ
=
G(g)
◦
η
A
2.2
Adjunctions
via
units
and
counits
53
for
any
g
:
F(A)
→
B,
and
f
¯
=
ε
B
◦
F(
f
)
for
any
f
:
A
→
G(B).
Proof
For
any
map
g
:
F(A)
→
B,
we
have
g
g
1
F(A)
−→
B
=
F(A)
−→
F(A)
−→
B
η
A
G(g)
=
A
−→
GF(A)
−→
G(B)
by
equation
(2.2),
giving
the
first
statement.
The
second
follows
by
duality.
Theorem
2.2.5
Take
categories
and
functors
A
o
/
B
.
There
is
a
one-to-
F
G
one
correspondence
between:
(a)
adjunctions
between
F
and
G
(with
F
on
the
left
and
G
on
the
right);
η
ε
(b)
pairs
1
A
−→
GF,
FG
−→
1
B
of
natural
transformations
satisfying
the
triangle
identities.
(Recall
that
by
definition,
an
adjunction
between
F
and
G
is
a
choice
of
isomorphism
(2.1)
for
each
A
and
B,
satisfying
the
naturality
equations
(2.2)
and
(2.3).)
Proof
We
have
shown
that
every
adjunction
between
F
and
G
gives
rise
to
a
pair
(η,
ε)
satisfying
the
triangle
identities.
We
now
have
to
show
that
this
process
is
bijective.
So,
take
a
pair
(η,
ε)
of
natural
transformations
satisfying
the
triangle
identities.
We
must
show
that
there
is
a
unique
adjunction
between
F
and
G
with
unit
η
and
counit
ε.
Uniqueness
follows
from
Lemma
2.2.4.
For
existence,
take
natural
transfor-
mations
η
and
ε
as
in
(b).
For
each
A
and
B,
define
functions
B(F(A),
B)
A
(A,
G(B)),
(2.5)
both
denoted
by
a
bar,
as
follows.
Given
g
∈
B(F(A),
B),
put
ḡ
=
G(g)
◦
η
A
∈
A
(A,
G(B)).
Similarly,
in
the
opposite
direction,
put
f
¯
=
ε
B
◦
F(
f
).
I
claim
that
for
each
A
and
B,
the
two
functions
g
7→
ḡ
and
f
7→
f
¯
are
mutu-
ally
inverse.
Indeed,
given
a
map
g
:
F(A)
→
B
in
B,
we
have
a
commutative
diagram
F(A)
F(η
A
)
1
/
FGF(A)
$
FG(g)
ε
F(A)
F(A)
/
FG(B)
ε
B
g
/
B.
54
Adjoints
The
composite
map
from
F(A)
to
B
by
one
route
around
the
outside
of
the
diagram
is
ε
B
◦
FG(g)
◦
F(η
A
)
=
ε
B
◦
F(ḡ)
=
ḡ
¯
,
and
by
the
other
is
g◦1
=
g,
so
ḡ
¯
=
g.
Dually,
f
¯
=
f
for
any
map
f
:
A
→
G(B)
in
A
.
This
proves
the
claim.
It
is
straightforward
to
check
the
naturality
equations
(2.2)
and
(2.3).
The
functions
(2.5)
therefore
define
an
adjunction.
Finally,
its
unit
and
counit
are
η
and
ε,
since
the
component
of
the
unit
at
A
is
1
F(A)
=
G(1
F(A)
)
◦
η
A
=
1
◦
η
A
=
η
A
,
and
dually
for
the
counit.
Corollary
2.2.6
Take
categories
and
functors
A
o
F
G
/
B
.
Then
F
a
G
if
and
η
ε
only
if
there
exist
natural
transformations
1
−→
GF
and
FG
−→
1
satisfying
the
triangle
identities.
Example
2.2.7
An
adjunction
between
ordered
sets
consists
of
order-pre-
serving
maps
A
o
f
g
/
B
such
that
∀a
∈
A,
∀b
∈
B,
f
(a)
≤
b
⇐⇒
a
≤
g(b).
(2.6)
This
is
because
both
sides
of
the
isomorphism
(2.1)
in
the
definition
of
ad-
junction
are
sets
with
at
most
one
element,
so
they
are
isomorphic
if
and
only
if
they
are
both
empty
or
both
nonempty.
The
naturality
requirements
(2.2)
and
(2.3)
hold
automatically,
since
in
an
ordered
set,
any
two
maps
with
the
same
domain
and
codomain
are
equal.
p
/
/
D
are
order-preserving
maps
of
Recall
from
Example
1.3.9
that
if
C
q
ordered
sets
then
there
is
at
most
one
natural
transformation
from
p
to
q,
and
there
is
one
if
and
only
if
p(c)
≤
q(c)
for
all
c
∈
C.
The
unit
of
the
adjunction
above
is
the
statement
that
a
≤
g
f
(a)
for
all
a
∈
A,
and
the
counit
is
the
statement
that
f
g(b)
≤
b
for
all
b
∈
B.
The
triangle
identities
say
nothing,
since
they
assert
the
equality
of
two
maps
in
an
ordered
set
with
the
same
domain
and
codomain.
In
the
case
of
ordered
sets,
Corollary
2.2.6
states
that
condition
(2.6)
is
equivalent
to:
∀a
∈
A,
a
≤
g
f
(a)
and
∀b
∈
B,
f
g(b)
≤
b.
This
equivalence
can
also
be
proved
directly
(Exercise
2.2.10).
55
2.2
Adjunctions
via
units
and
counits
For
instance,
let
X
be
a
topological
space.
Take
the
set
C
(X)
of
closed
sub-
sets
of
X
and
the
set
P(X)
of
all
subsets
of
X,
both
ordered
by
⊆.
There
are
order-preserving
maps
P(X)
o
Cl
i
/
C
(X)
where
i
is
the
inclusion
map
and
Cl
is
closure.
This
is
an
adjunction,
with
Cl
left
adjoint
to
i,
as
witnessed
by
the
fact
that
Cl(A)
⊆
B
⇐⇒
A
⊆
B
for
all
A
⊆
X
and
closed
B
⊆
X.
An
equivalent
statement
is
that
A
⊆
Cl(A)
for
all
A
⊆
X
and
Cl(B)
⊆
B
for
all
closed
B
⊆
X.
Either
way,
we
see
that
the
topological
operation
of
closure
arises
as
an
adjoint
functor.
Remark
2.2.8
Theorem
2.2.5
states
that
an
adjunction
may
be
regarded
as
a
quadruple
(F,
G,
η,
ε)
of
functors
and
natural
transformations
satisfying
the
triangle
identities.
An
equivalence
(F,
G,
η,
ε)
of
categories
(as
in
Definition
1.3.15)
is
not
necessarily
an
adjunction.
It
is
true
that
F
is
left
adjoint
to
G
(Exercise
2.3.10),
but
η
and
ε
are
not
necessarily
the
unit
and
counit
(because
there
is
no
reason
why
they
should
satisfy
the
triangle
identities).
Remark
2.2.9
There
is
a
way
of
drawing
natural
transformations
that
makes
the
triangle
identities
intuitively
plausible.
Suppose,
for
instance,
that
we
have
categories
and
functors
F
1
F
3
F
2
F
4
G
1
A
−→
C
1
−→
C
2
−→
C
3
−→
B,
G
2
A
−→
D
1
−→
B
and
a
natural
transformation
α
:
F
4
F
3
F
2
F
1
→
G
2
G
1
.
We
usually
draw
α
like
this:
F
1
F
2
5
C
1
1
C
2
F
3
-
C
3
⇓
α
A
-
D
1
G
1
F
4
G
2
However,
we
can
also
draw
α
as
a
string
diagram:
F
1
F
2
F
3
α
G
1
G
2
F
4
)
1
B
56
Adjoints
There
is
nothing
special
about
4
and
2;
we
could
replace
them
by
any
natural
numbers
m
and
n.
If
m
=
0
then
A
=
B
and
the
domain
of
α
is
1
A
(keeping
in
mind
the
last
paragraph
of
Remark
1.1.2(b)).
In
that
case,
the
disk
labelled
α
has
no
strings
coming
into
the
top.
Similarly,
if
n
=
0
then
there
are
no
strings
coming
out
of
the
bottom.
Vertical
composition
of
natural
transformations
corresponds
to
joining
string
diagrams
together
vertically,
and
horizontal
composition
corresponds
to
put-
ting
them
side
by
side.
The
identity
on
a
functor
F
is
drawn
as
a
simple
string,
F
F
Now
let
us
apply
this
notation
to
adjunctions.
The
unit
and
counit
are
drawn
as
η
G
F
and
ε
G
F
The
triangle
identities
now
become
the
topologically
plausible
equations
F
G
F
G
η
η
=
G
and
=
F
ε
ε
F
F
G
G
In
both
equations,
the
right-hand
side
is
obtained
from
the
left
by
simply
pulling
the
string
straight.
Exercises
2.2.10
Let
A
o
f
g
/
B
be
order-preserving
maps
between
ordered
sets.
Prove
directly
that
the
following
conditions
are
equivalent:
(a)
for
all
a
∈
A
and
b
∈
B,
f
(a)
≤
b
⇐⇒
a
≤
g(b);
2.2
Adjunctions
via
units
and
counits
57
(b)
a
≤
g(
f
(a))
for
all
a
∈
A
and
f
(g(b))
≤
b
for
all
b
∈
B.
(Both
conditions
state
that
f
a
g;
see
Example
2.2.7.)
2.2.11
(a)
Let
A
o
F
⊥
/
B
be
an
adjunction
with
unit
η
and
counit
ε.
Write
G
Fix(GF)
for
the
full
subcategory
of
A
whose
objects
are
those
A
∈
A
such
that
η
A
is
an
isomorphism,
and
dually
Fix(FG)
⊆
B.
Prove
that
the
adjunction
(F,
G,
η,
ε)
restricts
to
an
equivalence
(F
0
,
G
0
,
η
0
,
ε
0
)
between
Fix(GF)
and
Fix(FG).
(b)
Part
(a)
shows
that
every
adjunction
restricts
to
an
equivalence
between
full
subcategories
in
a
canonical
way.
Take
some
examples
of
adjunctions
and
work
out
what
this
equivalence
is.
2.2.12
(a)
Show
that
for
any
adjunction,
the
right
adjoint
is
full
and
faithful
if
and
only
if
the
counit
is
an
isomorphism.
(b)
An
adjunction
satisfying
the
equivalent
conditions
of
part
(a)
is
called
a
reflection.
(Compare
Example
2.1.3(d).)
Of
the
examples
of
adjunctions
given
in
this
chapter,
which
are
reflections?
2.2.13
(a)
Let
f
:
K
→
L
be
a
map
of
sets,
and
denote
by
f
∗
:
P(L)
→
P(K)
the
map
sending
a
subset
S
of
L
to
its
inverse
image
f
−1
S
⊆
K.
Then
f
∗
is
order-preserving
with
respect
to
the
inclusion
orderings
on
P(K)
and
P(L),
and
so
can
be
seen
as
a
functor.
Find
left
and
right
ad-
joints
to
f
∗
.
(b)
Now
let
X
and
Y
be
sets,
and
write
p
:
X
×
Y
→
X
for
first
projection.
Regard
a
subset
S
of
X
as
a
predicate
S
(x)
in
one
variable
x
∈
X,
and
similarly
a
subset
R
of
X
×
Y
as
a
predicate
R(x,
y)
in
two
variables.
What,
in
terms
of
predicates,
are
the
left
and
right
adjoints
to
p
∗
?
For
each
of
the
adjunctions,
interpret
the
unit
and
counit
as
logical
implications.
(Hint:
the
left
adjoint
to
p
∗
is
often
written
as
∃
Y
,
and
the
right
adjoint
as
∀
Y
.)
2.2.14
Given
a
functor
F
:
A
→
B
and
a
category
S
,
there
is
a
functor
F
∗
:
[B,
S
]
→
[A
,
S
]
defined
on
objects
Y
∈
[B,
S
]
by
F
∗
(Y)
=
Y
◦
F
and
on
F
/
maps
α
by
F
∗
(α)
=
αF.
Show
that
any
adjunction
A
o
⊥
B
and
category
G
S
give
rise
to
an
adjunction
[A
,
S
]
o
(Hint:
use
Theorem
2.2.5.)
G
∗
⊥
F
∗
/
[B,
S
]
.
58
Adjoints
2.3
Adjunctions
via
initial
objects
We
now
come
to
the
third
formulation
of
adjointness,
which
is
the
one
you
will
probably
see
most
often
in
everyday
mathematics.
Consider
once
more
the
adjunction
Vect
O
k
F
a
U
Set.
Let
S
be
a
set.
The
universal
property
of
F(S
),
the
vector
space
whose
basis
is
S
,
is
most
commonly
stated
like
this:
given
a
vector
space
V,
any
function
f
:
S
→
V
extends
uniquely
to
a
linear
map
f
¯
:
F(S
)
→
V.
As
remarked
in
Example
2.1.3(a),
forgetful
functors
are
often
forgotten:
in
this
statement,
‘
f
:
S
→
V’
should
strictly
speaking
be
‘
f
:
S
→
U(V)’.
Also,
the
word
‘extends’
refers
implicitly
to
the
embedding
η
S
:
→
UF(S
)
7→
s.
S
s
So
in
precise
language,
the
statement
reads:
for
any
V
∈
Vect
k
and
f
∈
Set(S
,
U(V)),
there
is
a
unique
f
¯
∈
Vect
k
(F(S
),
V)
such
that
the
diagram
η
S
S
/
U(F(S
))
U(
f
¯
)
#
U(V)
f
(2.7)
commutes.
(Compare
Example
0.4.)
In
this
section,
we
show
that
this
statement
is
equiv-
alent
to
the
statement
that
F
is
left
adjoint
to
U
with
unit
η.
To
do
this,
we
need
a
definition.
Definition
2.3.1
Given
categories
and
functors
B
Q
A
P
/
C
,
2.3
Adjunctions
via
initial
objects
59
the
comma
category
(P
⇒
Q)
(often
written
as
(P
↓
Q))
is
the
category
defined
as
follows:
•
objects
are
triples
(A,
h,
B)
with
A
∈
A
,
B
∈
B,
and
h
:
P(A)
→
Q(B)
in
C
;
•
maps
(A,
h,
B)
→
(A
0
,
h
0
,
B
0
)
are
pairs
(
f
:
A
→
A
0
,
g
:
B
→
B
0
)
of
maps
such
that
the
square
P(
f
)
P(A)
/
P(A
0
)
h
0
h
Q(B)
Q(g)
/
Q(B
0
)
commutes.
Remark
2.3.2
Given
A
,
B,
C
,
P
and
Q
as
above,
there
are
canonical
func-
tors
and
a
canonical
natural
transformation
as
shown:
/
B
A
⇒
(P
⇒
Q)
/
C
P
Q
In
a
suitable
2-categorical
sense,
(P
⇒
Q)
is
universal
with
this
property.
Example
2.3.3
Let
A
be
a
category
and
A
∈
A
.
The
slice
category
of
A
over
A,
denoted
by
A
/A,
is
the
category
whose
objects
are
maps
into
A
and
whose
maps
are
commutative
triangles.
More
precisely,
an
object
is
a
pair
(X,
h)
with
X
∈
A
and
h
:
X
→
A
in
A
,
and
a
map
(X,
h)
→
(X
0
,
h
0
)
in
A
/A
is
a
map
f
:
X
→
X
0
in
A
making
the
triangle
/
X
0
f
X
h
A
~
h
0
commute.
Slice
categories
are
a
special
case
of
comma
categories.
Recall
from
Exam-
ple
2.1.9
that
functors
1
→
A
are
just
objects
of
A
.
Now,
given
an
object
A
of
A
,
consider
the
comma
category
(1
A
⇒
A),
as
in
the
diagram
1
A
A
1
A
/
A
.
60
Adjoints
An
object
of
(1
A
⇒
A)
is
in
principle
a
triple
(X,
h,
B)
with
X
∈
A
,
B
∈
1,
and
h
:
X
→
A
in
A
;
but
1
has
only
one
object,
so
it
is
essentially
just
a
pair
(X,
h).
Hence
the
comma
category
(1
A
⇒
A)
has
the
same
objects
as
the
slice
category
A
/A.
One
can
check
that
it
has
the
same
maps
too,
so
that
A
/A
(1
A
⇒
A).
Dually
(reversing
all
the
arrows),
there
is
a
coslice
category
A/A
(A
⇒
1
A
),
whose
objects
are
the
maps
out
of
A.
Example
2.3.4
Let
G
:
B
→
A
be
a
functor
and
let
A
∈
A
.
We
can
form
the
comma
category
(A
⇒
G),
as
in
the
diagram
B
G
1
A
/
A
.
Its
objects
are
pairs
(B
∈
B,
f
:
A
→
G(B)).
A
map
(B,
f
)
→
(B
0
,
f
0
)
in
(A
⇒
G)
is
a
map
q
:
B
→
B
0
in
B
making
the
triangle
A
f
/
G(B)
f
0
!
G(B
0
)
G(q)
commute.
Notice
how
this
diagram
resembles
the
diagram
(2.7)
in
the
vector
space
ex-
ample.
We
will
use
comma
categories
(A
⇒
G)
to
capture
the
kind
of
universal
property
discussed
there.
Speaking
casually,
we
say
that
f
:
A
→
G(B)
is
an
object
of
(A
⇒
G),
when
what
we
should
really
say
is
that
the
pair
(B,
f
)
is
an
object
of
(A
⇒
G).
There
is
potential
for
confusion
here,
since
there
may
be
different
objects
B,
B
0
of
B
with
G(B)
=
G(B
0
).
Nevertheless,
we
will
often
use
this
convention.
We
now
make
the
connection
between
comma
categories
and
adjunctions.
Lemma
2.3.5
Take
an
adjunction
A
o
F
⊥
G
/
B
and
an
object
A
∈
A
.
Then
the
unit
map
η
A
:
A
→
GF(A)
is
an
initial
object
of
(A
⇒
G).
Proof
Let
(B,
f
:
A
→
G(B))
be
an
object
of
(A
⇒
G).
We
have
to
show
that
there
is
exactly
one
map
from
(F(A),
η
A
)
to
(B,
f
).
A
map
(F(A),
η
A
)
→
(B,
f
)
in
(A
⇒
G)
is
a
map
q
:
F(A)
→
B
in
B
such
61
2.3
Adjunctions
via
initial
objects
that
A
η
A
/
GF(A)
f
"
G(B)
(2.8)
G(q)
commutes.
But
G(q)
◦
η
A
=
q̄
by
Lemma
2.2.4,
so
(2.8)
commutes
if
and
only
if
f
=
q̄,
if
and
only
if
q
=
f
¯
.
Hence
f
¯
is
the
unique
map
(F(A),
η
A
)
→
(B,
f
)
in
(A
⇒
G).
We
now
meet
our
third
and
final
formulation
of
adjointness.
Theorem
2.3.6
F
Take
categories
and
functors
A
o
G
/
B
.
There
is
a
one-to-
one
correspondence
between:
(a)
adjunctions
between
F
and
G
(with
F
on
the
left
and
G
on
the
right);
(b)
natural
transformations
η
:
1
A
→
GF
such
that
η
A
:
A
→
GF(A)
is
initial
in
(A
⇒
G)
for
every
A
∈
A
.
Proof
We
have
just
shown
that
every
adjunction
between
F
and
G
gives
rise
to
a
natural
transformation
η
with
the
property
stated
in
(b).
To
prove
the
theo-
rem,
we
have
to
show
that
every
η
with
the
property
in
(b)
is
the
unit
of
exactly
one
adjunction
between
F
and
G.
By
Theorem
2.2.5,
an
adjunction
between
F
and
G
amounts
to
a
pair
(η,
ε)
of
natural
transformations
satisfying
the
triangle
identities.
So
it
is
enough
to
prove
that
for
every
η
with
the
property
in
(b),
there
exists
a
unique
natural
transformation
ε
:
FG
→
1
B
such
that
the
pair
(η,
ε)
satisfies
the
triangle
iden-
tities.
Let
η
:
1
A
→
GF
be
a
natural
transformation
with
the
property
in
(b).
Uniqueness
Suppose
that
ε,
ε
0
:
FG
→
1
B
are
natural
transformations
such
that
both
(η,
ε)
and
(η,
ε
0
)
satisfy
the
triangle
identities.
One
of
the
triangle
identities
states
that
for
all
B
∈
B,
the
triangle
G(B)
η
G(B)
1
commutes.
Thus,
ε
B
is
a
map
η
G(B)
FG(B),
G(B)
−→
G(FG(B))
/
G(FG(B))
&
(2.9)
G(ε
B
)
G(B)
−→
1
B,
G(B)
−→
G(B)
62
Adjoints
in
(G(B)
⇒
G).
The
same
is
true
of
ε
0
B
.
But
η
G(B)
is
initial,
so
there
is
only
one
such
map,
so
ε
B
=
ε
0
B
.
This
holds
for
all
B,
so
ε
=
ε
0
.
Existence
For
B
∈
B,
define
ε
B
:
FG(B)
→
B
to
be
the
unique
map
FG(B),
η
G(B)
→
B,
1
G(B)
in
(G(B)
⇒
G).
(So
by
definition
of
ε
B
,
triangle
(2.9)
commutes.)
We
show
that
(ε
B
)
B∈B
is
a
natural
transformation
FG
→
1
such
that
η
and
ε
satisfy
the
triangle
identities.
q
To
prove
naturality,
take
B
−→
B
0
in
B.
We
have
commutative
diagrams
η
G(B)
G(B)
1
G(q)
/
GFG(B)
%
η
G(B)
G(B)
/
GFG(B)
G(q)
G(ε
B
)
G(B
0
)
G(B)
GFG(q)
η
G(B
0
)
G(q)
G(B
0
)
1
G(q)
/
GFG(B
0
)
G(ε
B
0
)
%
0
3
G(B
).
So
q
◦
ε
B
and
ε
B
0
◦
FG(q)
are
both
maps
η
G(B)
→
G(q)
in
(G(B)
⇒G),
and
since
η
G(B)
is
initial,
they
must
be
equal.
This
proves
naturality
of
ε
with
respect
to
q.
Hence
ε
is
a
natural
transformation.
We
have
already
observed
that
one
of
the
triangle
identities,
equation
(2.9),
holds.
The
other
states
that
for
A
∈
A
,
F(A)
F(η
A
)
/
FGF(A)
%
1
F(A)
ε
F(A)
F(A)
commutes.
To
prove
it,
we
repeat
our
previous
technique:
there
are
commuta-
tive
diagrams
A
η
A
/
GF(A)
A
η
A
η
A
η
A
GF(A)
G(1
F(A)
)
GF(η
A
)
η
GF(A)
1
GF(A)
η
A
/
GF(A)
/
GFGF(A)
G(ε
F(A)
)
&
4
GF(A),
2.3
Adjunctions
via
initial
objects
so
by
initiality
of
η
A
,
we
have
ε
F(A)
◦
F(η
A
)
=
1
F(A)
,
as
required.
63
In
Section
6.3
we
will
meet
the
adjoint
functor
theorems,
which
state
condi-
tions
under
which
a
functor
is
guaranteed
to
have
a
left
adjoint.
The
following
corollary
is
the
starting
point
for
their
proofs.
Corollary
2.3.7
Let
G
:
B
→
A
be
a
functor.
Then
G
has
a
left
adjoint
if
and
only
if
for
each
A
∈
A
,
the
category
(A
⇒
G)
has
an
initial
object.
Proof
Lemma
2.3.5
proves
‘only
if’.
To
prove
‘if’,
let
us
choose
for
each
A
∈
A
an
initial
object
of
(A
⇒
G)
and
call
it
F(A),
η
A
:
A
→
GF(A)
.
(Here
F(A)
and
η
A
are
just
the
names
we
choose
to
use.)
For
each
map
f
:
A
→
A
0
in
A
,
let
F(
f
)
:
F(A)
→
F(A
0
)
be
the
unique
map
such
that
η
A
A
f
%
/
G(F(A))
A
0
G(F(
f
))
η
A
0
(
G(F(A
0
))
commutes
(in
other
words,
the
unique
map
η
A
→
η
A
0
◦
f
in
(A
⇒
G)).
It
is
easily
checked
that
F
is
a
functor
A
→
B,
and
the
diagram
tells
us
that
η
is
a
natural
transformation
1
→
GF.
So
by
Theorem
2.3.6,
F
is
left
adjoint
to
G.
This
corollary
justifies
the
claim
made
at
the
beginning
of
the
section:
that
given
functors
F
and
G,
to
have
an
adjunction
F
a
G
amounts
to
having
maps
η
A
:
A
→
GF(A)
with
the
universal
property
stated
there.
Exercises
2.3.8
What
can
be
said
about
adjunctions
between
groups
(regarded
as
one-
object
categories)?
2.3.9
State
the
dual
of
Corollary
2.3.7.
How
would
you
prove
your
dual
state-
ment?
2.3.10
Let
(F,
G,
η,
ε)
be
an
equivalence
of
categories,
as
in
Definition
1.3.15.
Prove
that
F
is
left
adjoint
to
G
(heeding
the
warning
in
Remark
2.2.8).
2.3.11
Let
A
o
U
>
/
Set
be
an
adjunction.
Suppose
that
for
at
least
one
A
∈
F
A
,
the
set
U(A)
has
at
least
two
elements.
Prove
that
for
each
set
S
,
the
unit
map
η
S
:
S
→
UF(S
)
is
injective.
What
does
this
mean
in
the
case
of
the
usual
adjunction
between
Grp
and
Set?
64
Adjoints
2.3.12
Given
sets
A
and
B,
a
partial
function
from
A
to
B
is
a
pair
(S
,
f
)
consisting
of
a
subset
S
⊆
A
and
a
function
S
→
B.
(Think
of
it
as
like
a
function
from
A
to
B,
but
undefined
at
certain
elements
of
A.)
Let
Par
be
the
category
of
sets
and
partial
functions.
Show
that
Par
is
equivalent
to
Set
∗
,
the
category
of
sets
equipped
with
a
distinguished
element
and
functions
preserving
distinguished
elements.
Show
also
that
Set
∗
can
be
described
as
a
coslice
category
in
a
simple
way.
3
Interlude
on
sets
Sets
and
functions
are
ubiquitous
in
mathematics.
You
might
have
the
impres-
sion
that
they
are
most
strongly
connected
with
the
pure
end
of
the
subject,
but
this
is
an
illusion:
think
of
probability
density
functions
in
statistics,
data
sets
in
experimental
science,
planetary
motion
in
astronomy,
or
flow
in
fluid
dynamics.
Category
theory
is
often
used
to
shed
light
on
common
constructions
and
patterns
in
mathematics.
If
we
hope
to
do
this
in
an
advanced
context,
we
must
begin
by
settling
the
basic
notions
of
set
and
function.
That
is
the
purpose
of
the
first
section
of
this
chapter.
The
definition
of
category
mentions
a
‘collection’
of
objects
and
‘collec-
tions’
of
maps.
We
will
see
in
the
second
section
that
some
collections
are
too
big
to
be
sets,
which
leads
to
a
distinction
between
‘small’
and
‘large’
collec-
tions.
This
distinction
will
be
needed
later,
most
prominently
for
the
adjoint
functor
theorems
(Chapter
6).
The
final
section
takes
a
historical
look
at
set
theory.
It
also
explains
why
the
approach
to
sets
taken
in
this
chapter
is
more
relevant
to
most
of
mathematics
than
the
traditional
approach
is.
None
of
this
section
is
logically
necessary
for
anything
that
follows,
but
it
may
provide
useful
perspective.
I
do
not
assume
that
you
have
encountered
axiomatic
set
theory
of
any
kind.
If
you
have,
it
is
probably
best
to
put
it
out
of
your
mind
while
reading
this
chapter,
as
the
approach
to
set
theory
that
we
take
is
quite
different
from
the
approach
that
you
are
most
likely
to
be
familiar
with.
A
brief
comparison
of
the
traditional
and
categorical
approaches
can
be
found
at
the
very
end
of
the
chapter.
65
66
Interlude
on
sets
3.1
Constructions
with
sets
We
have
made
no
definition
of
‘set’,
nor
of
‘function’.
Nevertheless,
guided
by
our
intuition,
we
can
list
some
properties
that
we
expect
the
world
of
sets
and
functions
to
have.
For
instance,
we
can
describe
some
of
the
sets
that
we
think
ought
to
exist,
and
some
ways
of
building
new
sets
from
old.
Intuitively,
a
set
is
a
bag
of
points:
(There
may,
of
course,
be
infinitely
many.)
These
points,
or
elements,
are
not
related
to
one
another
in
any
way.
They
are
not
in
any
order,
they
do
not
come
with
any
algebraic
structure
(for
instance,
there
is
no
specified
way
of
multi-
plying
elements
together),
and
there
is
no
sense
of
what
it
means
for
one
point
to
be
close
to
another.
In
particular
examples,
we
might
have
some
extra
struc-
ture
in
mind;
for
instance,
we
often
equip
the
set
of
real
numbers
with
an
order,
a
field
structure
and
a
metric.
But
to
view
R
as
a
mere
set
is
to
ignore
all
that
structure,
to
regard
it
as
no
more
than
a
bunch
of
featureless
points.
Intuitively,
a
function
A
→
B
is
an
assignment
of
a
point
in
bag
B
to
each
point
in
bag
A:
We
can
do
one
function
after
another:
given
functions
we
obtain
a
composite
function
This
composition
of
functions
is
associative:
h
◦
(g
◦
f
)
=
(h
◦
g)
◦
f
.
There
is
also
an
identity
function
on
every
set.
Hence:
3.1
Constructions
with
sets
67
Sets
and
functions
form
a
category,
denoted
by
Set.
This
does
not
pin
things
down
much:
there
are
many
categories,
mostly
quite
unlike
the
category
of
sets.
So,
let
us
list
some
of
the
special
features
of
the
category
of
sets.
The
empty
set
There
is
a
set
∅
with
no
elements.
Suppose
that
someone
hands
you
a
pair
of
sets,
A
and
B,
and
tells
you
to
specify
a
function
from
A
to
B.
Then
your
task
is
to
specify
for
each
element
of
A
an
element
of
B.
The
larger
A
is,
the
longer
the
task;
the
smaller
A
is,
the
shorter
the
task.
In
particular,
if
A
is
empty
then
the
task
takes
no
time
at
all;
we
have
nothing
to
do.
So
there
is
a
function
from
∅
to
B
specified
by
doing
nothing.
On
the
other
hand,
there
cannot
be
two
different
ways
to
do
nothing,
so
there
is
only
one
function
from
∅
to
B.
Hence:
∅
is
an
initial
object
of
Set.
In
case
this
argument
seems
unconvincing,
here
is
an
alternative.
Suppose
that
we
have
a
set
A
with
disjoint
subsets
A
1
and
A
2
such
that
A
1
∪
A
2
=
A.
Then
a
function
from
A
to
B
amounts
to
a
function
from
A
1
to
B
together
with
a
function
from
A
2
to
B.
So
if
all
the
sets
are
finite,
we
should
have
the
rule
(number
of
functions
from
A
to
B)
=
(number
of
functions
from
A
1
to
B)
×
(number
of
functions
from
A
2
to
B).
In
particular,
we
could
take
A
1
=
A
and
A
2
=
∅.
This
would
force
the
number
of
functions
from
∅
to
B
to
be
1.
So
if
we
want
this
rule
to
hold
(and
surely
we
do!),
we
had
better
say
that
there
is
exactly
one
function
from
∅
to
B.
What
about
functions
into
∅?
There
is
exactly
one
function
∅
→
∅,
namely,
the
identity.
This
is
a
special
case
of
the
initiality
of
∅.
On
the
other
hand,
for
a
set
A
that
is
not
empty,
there
are
no
functions
A
→
∅,
because
there
is
nowhere
for
elements
of
A
to
go.
The
one-element
set
There
is
a
set
1
with
exactly
one
element.
For
any
set
A,
there
is
exactly
one
function
from
A
to
1,
since
every
element
of
A
must
be
mapped
to
the
unique
element
of
1.
That
is:
1
is
a
terminal
object
of
Set.
A
function
from
1
to
a
set
B
is
just
a
choice
of
an
element
of
B.
In
short,
the
functions
1
→
B
are
the
elements
of
B.
Hence:
The
concept
of
element
is
a
special
case
of
the
concept
of
function.
68
Interlude
on
sets
Products
Any
two
sets
A
and
B
have
a
product,
A
×
B.
Its
elements
are
the
ordered
pairs
(a,
b)
with
a
∈
A
and
b
∈
B.
Ordered
pairs
are
familiar
from
coordinate
geometry.
All
that
matters
about
them
is
that
for
a,
a
0
∈
A
and
b,
b
0
∈
B,
(a,
b)
=
(a
0
,
b
0
)
⇐⇒
a
=
a
0
and
b
=
b
0
.
More
generally,
take
any
set
I
and
any
family
(A
i
)
i∈I
of
sets.
There
is
a
product
Q
set
i∈I
A
i
,
whose
elements
are
families
(a
i
)
i∈I
with
a
i
∈
A
i
for
each
i.
Just
as
for
ordered
pairs,
(a
i
)
i∈I
=
(a
0
i
)
i∈I
⇐⇒
a
i
=
a
0
i
for
all
i
∈
I.
Sums
Any
two
sets
A
and
B
have
a
sum
A
+
B.
Thinking
of
sets
as
bags
of
points,
the
sum
of
two
sets
is
obtained
by
putting
all
the
points
into
one
big
bag:
+
=
If
A
and
B
are
finite
sets
with
m
and
n
elements
respectively,
then
A
+
B
always
has
m
+
n
elements.
It
makes
no
difference
what
the
elements
of
A
+
B
are
called;
as
usual,
we
only
care
what
A
+
B
is
up
to
isomorphism.
There
are
inclusion
functions
i
j
A
−→
A
+
B
←−
B
such
that
the
union
of
the
images
of
i
and
j
is
all
of
A
+
B
and
the
intersection
of
the
images
is
empty.
Sum
is
sometimes
called
disjoint
union
and
written
as
q.
It
is
not
to
be
confused
with
(ordinary)
union
∪.
For
a
start,
we
can
take
the
sum
of
any
two
sets
A
and
B,
whereas
A
∪
B
only
really
makes
sense
when
A
and
B
come
as
subsets
of
some
larger
set.
(For
to
say
what
A
∪
B
is,
we
need
to
know
which
elements
of
A
are
equal
to
which
elements
of
B.)
And
even
if
A
and
B
do
come
as
subsets
of
some
larger
set,
A
+
B
and
A
∪
B
can
be
different.
For
example,
take
the
subsets
A
=
{1,
2,
3}
and
B
=
{3,
4}
of
N.
Then
A
∪
B
has
4
elements,
but
A
+
B
has
3
+
2
=
5
elements.
P
More
generally,
any
family
(A
i
)
i∈I
of
sets
has
a
sum
i∈I
A
i
.
If
I
is
finite
and
P
P
each
A
i
is
finite,
say
with
m
i
elements,
then
i∈I
A
i
has
i∈I
m
i
elements.
3.1
Constructions
with
sets
69
Sets
of
functions
For
any
two
sets
A
and
B,
we
can
form
the
set
A
B
of
func-
tions
from
B
to
A.
Q
This
is
a
special
case
of
the
product
construction:
A
B
is
the
product
b∈B
A
Q
of
the
constant
family
(A)
b∈B
.
Indeed,
an
element
of
b∈B
A
is
a
family
(a
b
)
b∈B
consisting
of
one
element
a
b
∈
A
for
each
b
∈
B;
in
other
words,
it
is
a
function
B
→
A.
Digression
on
arithmetic
We
are
using
notation
reminiscent
of
arithmetic:
A
×
B,
A
+
B,
and
A
B
.
There
is
good
reason
for
this:
if
A
is
a
finite
set
with
m
elements
and
B
a
finite
set
with
n
elements,
then
A×
B
has
m×n
elements,
A+
B
has
m
+
n
elements,
and
A
B
has
m
n
elements.
Our
notation
1
for
a
one-element
set
and
the
alternative
notation
0
for
the
empty
set
∅
also
follow
this
pattern.
All
the
usual
laws
of
arithmetic
have
their
counterparts
for
sets:
A
×
(B
+
C)
(A
×
B)
+
(A
×
C),
A
B+C
A
B
×
A
C
,
(A
B
)
C
A
B×C
,
and
so
on,
where
is
isomorphism
in
the
category
of
sets.
(For
the
last
one,
see
Example
2.1.6.)
These
isomorphisms
hold
for
all
sets,
not
just
finite
ones.
The
two-element
set
Let
2
be
the
set
1
+
1
(a
set
with
two
elements!).
For
reasons
that
will
soon
become
clear,
I
will
write
the
elements
of
2
as
true
and
false.
Let
A
be
a
set.
Given
a
subset
S
of
A,
we
obtain
a
function
χ
S
:
A
→
2
(the
characteristic
function
of
S
⊆
A),
where
if
a
∈
S
,
true
χ
S
(a)
=
false
if
a
<
S
(a
∈
A).
Conversely,
given
a
function
f
:
A
→
2,
we
obtain
a
subset
f
−1
{true}
=
{a
∈
A
|
f
(a)
=
true}
of
A.
These
two
processes
are
mutually
inverse;
that
is,
χ
S
is
the
unique
func-
tion
f
:
A
→
2
such
that
f
−1
{true}
=
S
.
Hence:
Subsets
of
A
correspond
one-to-one
with
functions
A
→
2.
We
already
know
that
the
functions
from
A
to
2
form
a
set,
2
A
.
When
we
are
thinking
of
2
A
as
the
set
of
all
subsets
of
A,
we
call
it
the
power
set
of
A
and
write
it
as
P(A).
70
Interlude
on
sets
Equalizers
It
would
be
nice
if,
given
a
set
A,
we
could
define
a
subset
S
of
A
by
specifying
a
property
that
the
elements
of
S
are
to
satisfy:
S
=
{a
∈
A
|
some
property
of
a
holds}.
It
is
hard
to
give
a
general
definition
of
‘property’.
There
is,
however,
a
special
type
of
property
that
is
easy
to
handle:
equality
of
two
functions.
Precisely,
f
given
sets
and
functions
A
g
/
/
B
,
there
is
a
set
{a
∈
A
|
f
(a)
=
g(a)}.
This
set
is
called
the
equalizer
of
f
and
g,
since
it
is
the
part
of
A
on
which
the
two
functions
are
equal.
Quotients
You
are
probably
familiar
with
quotient
groups
and
quotient
rings
(sometimes
called
factor
groups
and
factor
rings)
in
algebra.
Quotients
also
come
up
everywhere
in
topology,
such
as
when
we
glue
together
opposite
sides
of
a
square
to
make
a
cylinder.
But
the
most
basic
context
for
quotients
is
that
of
sets.
Let
A
be
a
set
and
∼
an
equivalence
relation
on
A.
There
is
a
set
A/∼,
the
quotient
of
A
by
∼,
whose
elements
are
the
equivalence
classes.
For
example,
given
a
group
G
and
a
normal
subgroup
N,
define
an
equivalence
relation
∼
on
G
by
g
∼
h
⇐⇒
gh
−1
∈
N;
then
G/∼
=
G/N.
There
is
also
a
canonical
map
p
:
A
→
A/∼,
sending
an
element
of
A
to
its
equivalence
class.
It
is
surjective,
and
has
the
property
that
p(a)
=
p(a
0
)
⇐⇒
a
∼
a
0
.
In
fact,
it
has
a
universal
property:
any
function
f
:
A
→
B
such
that
a
∼
a
0
=⇒
f
(a)
=
f
(a
0
)
∀a,
a
0
∈
A,
(3.1)
factorizes
uniquely
through
p,
as
in
the
diagram
A
p
/
A/∼
f
!
B.
f
¯
Thus,
for
any
set
B,
the
functions
A/∼
→
B
correspond
one-to-one
with
the
functions
f
:
A
→
B
satisfying
(3.1).
This
fact
is
at
the
heart
of
the
famous
isomorphism
theorems
of
algebra.
3.1
Constructions
with
sets
71
We
have
now
listed
the
properties
of
sets
and
functions
that
will
be
most
important
for
us.
Here
are
two
more.
Natural
numbers
A
function
with
domain
N
is
usually
called
a
sequence.
A
crucial
property
of
N
is
that
sequences
can
be
defined
recursively:
given
a
set
X,
an
element
a
∈
X,
and
a
function
r
:
X
→
X,
there
is
a
unique
sequence
(x
n
)
∞
n=0
of
elements
of
X
such
that
x
0
=
a,
x
n+1
=
r(x
n
)
for
all
n
∈
N.
This
property
refers
to
two
pieces
of
structure
on
N:
the
element
0,
and
the
function
s
:
N
→
N
defined
by
s(n)
=
n+1.
Reformulated
in
terms
of
functions,
and
writing
x
n
=
x(n),
the
property
is
this:
for
any
set
X,
element
a
∈
X,
and
function
r
:
X
→
X,
there
is
a
unique
function
x
:
N
→
X
such
that
x(0)
=
a
and
x◦s
=
r◦x.
Exercise
3.1.2
asks
you
to
show
that
this
is
a
universal
property
of
N,
0
and
s.
Choice
Let
f
:
A
→
B
be
a
map
in
a
category
A
.
A
section
(or
right
inverse)
of
f
is
a
map
i
:
B
→
A
in
A
such
that
f
◦
i
=
1
B
.
In
the
category
of
sets,
any
map
with
a
section
is
certainly
surjective.
The
converse
statement
is
called
the
axiom
of
choice:
Every
surjection
has
a
section.
It
is
called
‘choice’
because
specifying
a
section
of
f
:
A
→
B
amounts
to
choosing,
for
each
b
∈
B,
an
element
of
the
nonempty
set
{a
∈
A
|
f
(a)
=
b}.
The
properties
listed
above
are
not
theorems,
since
we
do
not
have
rigorous
definitions
of
set
and
function.
What,
then,
is
their
status?
Definitions
in
mathematics
usually
depend
on
previous
definitions.
A
vector
space
is
defined
as
an
abelian
group
with
a
scalar
multiplication.
An
abelian
group
is
defined
as
a
group
with
a
certain
property.
A
group
is
defined
as
a
set
with
certain
extra
structure.
A
set
is
defined
as.
.
.
well,
what?
We
cannot
keep
going
back
indefinitely,
otherwise
we
quite
literally
would
not
know
what
we
were
talking
about.
We
have
to
start
somewhere.
In
other
words,
there
have
to
be
some
basic
concepts
not
defined
in
terms
of
anything
else.
The
concept
of
set
is
usually
taken
to
be
one
of
the
basic
ones,
which
is
why
you
have
probably
never
read
a
sentence
beginning
‘Definition:
A
set
is.
.
.
’.
We
will
treat
function
as
a
basic
concept,
too.
But
now
there
seems
to
be
a
problem.
If
these
basic
concepts
are
not
defined
in
terms
of
anything
else,
how
are
we
to
know
what
they
really
are?
How
are
we
going
to
reason
in
the
watertight,
logical
way
upon
which
mathematics
72
Interlude
on
sets
depends?
We
cannot
simply
trust
our
intuitions,
since
your
intuitive
idea
of
set
might
be
slightly
different
from
mine,
and
if
it
came
to
a
dispute
about
how
sets
behave,
we
would
have
no
way
of
deciding
who
was
right.
The
problem
is
solved
as
follows.
Instead
of
defining
a
set
to
be
a
such-and-
such
and
a
function
to
be
a
such-and-such
else,
we
list
some
properties
that
we
assume
sets
and
functions
to
have.
In
other
words,
we
never
attempt
to
say
what
sets
and
functions
are;
we
just
say
what
you
can
do
with
them.
In
his
excellent
book
Mathematics:
A
Very
Short
Introduction,
Timothy
Gowers
(2002)
considers
the
question:
‘What
is
the
black
king
in
chess?’
He
swiftly
points
out
that
this
question
is
rather
peculiar.
It
is
not
important
that
the
black
king
is
a
small
piece
of
wood,
painted
a
certain
colour
and
carved
into
a
certain
shape.
We
could
equally
well
use
a
scrap
of
paper
with
‘BK’
written
on
it.
What
matters
is
what
the
black
king
does:
it
can
move
in
certain
ways
but
not
others,
according
to
the
rules
of
chess.
Similarly,
we
might
not
be
able
to
say
directly
what
a
set
or
function
‘is’,
but
we
agree
that
they
are
to
satisfy
all
the
properties
on
the
list.
So
the
list
of
properties
acts
as
an
agreement
on
how
to
use
the
words
‘set’
and
‘function’,
just
as
the
rules
of
chess
act
as
an
agreement
on
how
to
use
the
chess
pieces.
What
we
are
doing
is
often
referred
to
as
foundations.
In
this
metaphor,
the
foundation
consists
of
the
basic
concepts
(set
and
function),
which
are
not
built
on
anything
else,
but
are
assumed
to
satisfy
a
stated
list
of
properties.
On
top
of
the
foundations
are
built
some
basic
definitions
and
theorems.
On
top
of
those
are
built
further
definitions
and
theorems,
and
so
on,
towering
upwards.
The
properties
above
are
stated
informally,
but
they
can
be
formalized
using
some
categorical
language.
(See
Lawvere
and
Rosebrugh
(2003)
or
Leinster
(2014).)
In
the
formal
version,
we
begin
by
saying
that
sets
and
functions
form
a
category,
Set.
We
then
list
some
properties
of
this
category.
For
example,
the
category
is
required
to
have
an
initial
and
a
terminal
object,
and
the
properties
described
informally
under
the
headings
‘Products’
and
‘Equalizers’
are
made
formal
by
the
statement
that
Set
‘has
limits’
(a
phrase
defined
in
Chapter
5).
While
we
were
making
the
list,
we
were
guided
by
our
intuition
about
sets.
But
once
it
is
made,
our
intuition
plays
no
further
official
role:
any
disputes
about
the
nature
of
sets
are
settled
by
consulting
the
list
of
properties.
(A
subtlety
arises.
Whatever
list
of
properties
one
writes
down,
there
might
be
some
questions
that
cannot
be
settled.
In
other
words,
there
might
be
multi-
ple
inequivalent
categories
satisfying
all
the
properties
listed.
This
gets
us
into
the
realm
of
advanced
logic:
Gödel
incompleteness,
the
continuum
hypothesis,
and
so
on,
all
beyond
the
scope
of
this
book.)
Now
let
us
look
again
at
the
section
on
the
empty
set.
You
might
have
felt
that
I
was
on
shaky
ground
when
trying
to
convince
you
that
∅
is
initial.
But
the
3.2
Small
and
large
categories
73
point
is
that
I
do
not
need
to
convince
you
that
this
is
a
true
statement;
I
only
need
to
convince
you
that
it
is
a
convenient
assumption.
Compare
the
rule
for
numbers
that
x
0
=
1.
One
can
reasonably
argue
that
0
copies
of
x
multiplied
together
ought
to
be
1,
but
really
the
best
justification
for
this
rule
is
conve-
nience:
it
makes
other
rules
such
as
x
m+n
=
x
m
·
x
n
true
without
exception.
Indeed,
it
does
not
even
make
sense
to
ask
whether
it
is
‘true’
that
∅
is
initial
until
we
have
written
down
our
assumptions
about
how
sets
and
functions
be-
have.
For
until
then,
what
could
‘true’
mean?
There
is
no
physical
world
of
sets
against
which
to
test
such
statements.
We
can
make
whatever
assumptions
about
sets
we
like,
but
some
lead
to
more
interesting
mathematics
than
others.
If,
for
instance,
you
want
to
assume
that
there
are
no
functions
from
∅
to
any
other
set,
you
can,
but
the
tower
of
mathematics
built
on
that
foundation
will
look
different
from
what
you
are
used
to,
and
probably
not
in
a
good
way.
For
example,
the
‘number
of
functions’
rule
(page
67)
will
fail,
and
there
will
be
further
unpleasant
surprises
higher
up
the
tower.
Exercises
3.1.1
The
diagonal
functor
∆
:
Set
→
Set
×
Set
is
defined
by
∆(A)
=
(A,
A)
for
all
sets
A.
Exhibit
left
and
right
adjoints
to
∆.
3.1.2
In
the
paragraph
headed
‘Natural
numbers’,
it
was
observed
that
the
set
N,
together
with
the
element
0
and
the
function
s
:
N
→
N,
has
a
certain
property.
This
property
can
be
understood
as
stating
that
the
triple
(N,
0,
s)
is
the
initial
object
of
a
certain
category
C
.
Find
C
.
3.2
Small
and
large
categories
We
have
now
made
some
assumptions
about
the
nature
of
sets.
One
conse-
quence
of
those
assumptions
is
that
in
many
of
the
categories
we
have
met,
the
collection
of
all
objects
is
too
large
to
form
a
set.
In
fact,
even
the
collection
of
isomorphism
classes
of
objects
is
often
too
large
to
form
a
set.
In
this
section,
I
will
explain
what
these
statements
mean,
and
prove
them.
This
section
is
not
of
central
importance.
As
this
book
proceeds,
I
will
say
as
little
as
possible
about
the
distinction
between
sets
and
collections
too
large
to
be
sets.
Nevertheless,
the
distinction
begins
to
matter
in
parts
of
category
theory
lying
just
within
the
scope
of
this
book
(the
adjoint
functor
theorems),
as
well
as
beyond.
74
Interlude
on
sets
Given
sets
A
and
B,
write
|A|
≤
|B|
(or
|B|
≥
|A|)
if
there
exists
an
injection
A
→
B.
We
give
no
meaning
to
the
expression
‘|A|’
or
‘|B|’
in
isolation.
(It
would
perhaps
be
more
logical
to
write
A
≤
B
rather
than
|A|
≤
|B|,
but
the
notation
is
well-established.)
In
the
case
of
finite
sets,
it
just
means
that
the
number
of
elements
of
A
is
less
than
or
equal
to
the
number
of
elements
of
B.
Since
identity
maps
are
injective,
|A|
≤
|A|
for
all
sets
A,
and
since
the
com-
posite
of
two
injections
is
an
injection,
|A|
≤
|B|
≤
|C|
=⇒
|A|
≤
|C|
.
Also,
if
A
B
then
|A|
≤
|B|
≤
|A|.
Less
obvious
is
the
converse:
Theorem
3.2.1
(Cantor–Bernstein)
then
A
B.
Proof
Let
A
and
B
be
sets.
If
|A|
≤
|B|
≤
|A|
Exercise
3.2.12.
These
observations
tell
us
that
≤
is
a
preorder
(Example
1.1.8(e))
on
the
collection
of
all
sets.
It
is
not
a
genuine
order,
since
|A|
≤
|B|
≤
|A|
only
implies
that
A
B,
not
A
=
B.
We
write
|A|
=
|B|,
and
say
that
A
and
B
have
the
same
cardinality,
if
A
B,
or
equivalently
if
|A|
≤
|B|
≤
|A|.
As
long
as
we
do
not
confuse
equality
with
isomorphism,
the
sign
≤
behaves
as
we
might
imagine.
For
example,
write
|A|
<
|B|
if
|A|
≤
|B|
and
|A|
,
|B|.
Then
|A|
≤
|B|
<
|C|
=⇒
|A|
<
|C|
(3.2)
for
sets
A,
B
and
C.
Indeed,
we
have
already
established
that
|A|
≤
|C|,
and
the
strict
inequality
follows
from
Theorem
3.2.1.
Here
is
another
fundamental
result
of
set
theory.
Theorem
3.2.2
(Cantor)
Let
A
be
a
set.
Then
|A|
<
|P(A)|.
Recall
that
P(A)
is
the
power
set
of
A.
The
lemma
is
easy
for
finite
sets,
since
if
A
has
n
elements
then
P(A)
has
2
n
elements,
and
n
<
2
n
.
Proof
Exercise
3.2.13.
Corollary
3.2.3
For
every
set
A,
there
is
a
set
B
such
that
|A|
<
|B|.
In
other
words,
there
is
no
biggest
set.
We
now
justify
the
claim
made
at
the
beginning
of
this
section:
that
for
many
familiar
categories,
the
collection
of
isomorphism
classes
of
objects
is
too
large
to
form
a
set.
We
begin
by
doing
this
for
the
category
Set
itself.
As
a
clue
to
why
the
collection
of
isomorphism
classes
of
sets
might
be
too
3.2
Small
and
large
categories
75
large
to
form
a
set,
consider
the
following
statement:
the
collection
of
isomor-
phism
classes
of
finite
sets
is
too
large
to
form
a
finite
set.
This
is
because
there
is
one
isomorphism
class
of
finite
sets
for
each
natural
number,
but
there
are
infinitely
many
natural
numbers.
Proposition
3.2.4
Let
I
be
a
set,
and
let
(A
i
)
i∈I
be
a
family
of
sets.
Then
there
exists
a
set
not
isomorphic
to
any
of
the
sets
A
i
.
Proof
Put
A
=
P
X
!
A
i
,
i∈I
the
power
set
of
the
sum
of
the
sets
A
i
.
For
each
j
∈
I,
we
have
the
inclusion
P
function
A
j
→
i∈I
A
i
,
so
by
Theorem
3.2.2,
X
A
j
≤
A
i
<
|A|
.
i∈I
Hence
A
j
<
|A|
by
(3.2),
and
in
particular,
A
j
6
A.
We
use
the
word
class
informally
to
mean
any
collection
of
mathematical
objects.
All
sets
are
classes,
but
some
classes
(such
as
the
class
of
all
sets)
are
too
big
to
be
sets.
A
class
will
be
called
small
if
it
is
a
set,
and
large
otherwise.
For
example,
Proposition
3.2.4
states
that
the
class
of
isomorphism
classes
of
sets
is
large.
The
crucial
point
is:
Any
individual
set
is
small,
but
the
class
of
sets
is
large.
This
is
even
true
if
we
pretend
that
isomorphic
sets
are
equal.
Although
the
‘definition’
of
class
is
not
precise,
it
will
do
for
our
purposes.
We
make
a
naive
distinction
between
small
and
large
collections,
and
implicitly
use
some
intuitively
plausible
principles
(for
example,
that
any
subcollection
of
a
small
collection
is
small).
A
category
A
is
small
if
the
class
or
collection
of
all
maps
in
A
is
small,
and
large
otherwise.
If
A
is
small
then
the
class
of
objects
of
A
is
small
too,
since
objects
correspond
one-to-one
with
identity
maps.
A
category
A
is
locally
small
if
for
each
A,
B
∈
A
,
the
class
A
(A,
B)
is
small.
(So,
small
implies
locally
small.)
Many
authors
take
local
smallness
to
be
part
of
the
definition
of
category.
The
class
A
(A,
B)
is
often
called
the
hom-
set
from
A
to
B,
although
strictly
speaking,
we
should
only
call
it
this
when
A
is
locally
small.
Example
3.2.5
Set
is
locally
small,
because
for
any
two
sets
A
and
B,
the
functions
from
A
to
B
form
a
set.
This
was
one
of
the
properties
of
sets
stated
in
Section
3.1.
76
Interlude
on
sets
Example
3.2.6
Vect
k
,
Grp,
Ab,
Ring
and
Top
are
all
locally
small.
For
ex-
ample,
given
rings
A
and
B,
a
homomorphism
from
A
to
B
is
a
function
from
A
to
B
with
certain
properties,
and
the
collection
of
all
functions
from
A
to
B
is
small,
so
the
collection
of
homomorphisms
from
A
to
B
is
certainly
small.
A
category
is
small
if
and
only
if
it
is
locally
small
and
its
class
of
objects
is
small.
Again,
it
may
help
to
consider
a
similar
fact
about
finiteness:
a
category
A
is
finite
(that
is,
the
class
of
all
maps
in
A
is
finite)
if
and
only
if
it
is
locally
finite
(that
is,
each
class
A
(A,
B)
is
finite)
and
its
class
of
objects
is
finite.
Example
3.2.7
Consider
the
category
B
defined
in
the
last
paragraph
of
Ex-
ample
1.3.20.
Its
objects
correspond
to
the
natural
numbers,
which
form
a
set,
so
the
class
of
objects
of
B
is
small.
Each
hom-set
B(m,
n)
is
a
set
(indeed,
a
finite
set),
so
B
is
locally
small.
Hence
B
is
small.
A
category
is
essentially
small
if
it
is
equivalent
to
some
small
category.
For
example,
the
category
of
finite
sets
is
essentially
small,
since
by
Exam-
ple
1.3.20,
it
is
equivalent
to
the
small
category
B
just
mentioned.
If
two
categories
A
and
B
are
equivalent,
the
class
of
isomorphism
classes
of
objects
of
A
is
in
bijection
with
that
of
B.
In
a
small
category,
the
class
of
objects
is
small,
so
the
class
of
isomorphism
classes
of
objects
is
certainly
small.
Hence
in
an
essentially
small
category,
the
class
of
isomorphism
classes
of
objects
is
small.
From
this
we
deduce:
Proposition
3.2.8
Set
is
not
essentially
small.
Proof
Proposition
3.2.4
states
that
the
class
of
isomorphism
classes
of
sets
is
large.
The
result
follows.
By
adapting
this
argument,
we
can
show
that
many
of
our
standard
examples
of
categories
are
not
essentially
small.
The
strategy
is
to
prove
that
there
are
at
least
as
many
objects
of
our
category
as
there
are
sets.
Example
3.2.9
For
any
field
k,
the
category
Vect
k
of
vector
spaces
over
k
is
not
essentially
small.
As
in
the
proof
of
Proposition
3.2.8,
it
is
enough
to
prove
that
the
class
of
isomorphism
classes
of
vector
spaces
is
large.
In
other
words,
it
is
enough
to
prove
that
for
any
set
I
and
family
(V
i
)
i∈I
of
vector
spaces,
there
exists
a
vector
space
not
isomorphic
to
any
of
the
spaces
V
i
.
U
/
To
show
this,
write
Vect
k
o
>
Set
for
the
free
and
forgetful
functors.
As
in
F
the
proof
of
Proposition
3.2.4,
the
set
S
=
P
X
i∈I
!
U(V
i
)
3.2
Small
and
large
categories
77
has
the
property
that
|U(V
i
)|
<
|S
|
for
all
i
∈
I.
The
free
vector
space
F(S
)
on
S
contains
a
copy
of
S
as
a
basis,
so
|S
|
≤
|UF(S
)|.
Hence
|U(V
i
)|
<
|UF(S
)|
for
all
i,
and
so
F(S
)
6
V
i
for
all
i,
as
required.
Similarly,
none
of
the
categories
Grp,
Ab,
Ring
and
Top
is
essentially
small
(Exercise
3.2.14).
Recall
that
the
category
of
all
categories
and
functors
is
written
as
CAT.
Definition
3.2.10
We
denote
by
Cat
the
category
of
small
categories
and
functors
between
them.
Example
3.2.11
Monoids
are
by
definition
sets
equipped
with
certain
struc-
ture,
so
the
one-object
categories
that
they
correspond
to
are
small.
Let
M
be
the
full
subcategory
of
Cat
consisting
of
the
one-object
categories.
Then
there
is
an
equivalence
of
categories
Mon
'
M
.
This
is
proved
by
the
argument
in
Example
1.3.21,
noting
that
because
each
object
of
M
is
a
small
one-object
category,
the
collection
of
maps
from
the
single
object
to
itself
really
is
a
set.
Exercises
3.2.12
(a)
Let
A
be
a
set.
Let
θ
:
P(A)
→
P(A)
be
a
map
that
is
order-
preserving
with
respect
to
inclusion.
A
fixed
point
of
θ
is
an
element
S
∈
P(A)
such
that
θ(S
)
=
S
.
By
considering
[
S
=
R,
R∈P(A)
:
θ(R)⊇R
prove
that
θ
has
at
least
one
fixed
point.
(b)
Take
sets
and
functions
A
o
f
g
/
B
.
Using
(a),
show
that
there
is
some
sub-
set
S
of
A
such
that
g(B
\
f
S
)
=
A
\
S
.
(c)
Deduce
the
Cantor–Bernstein
theorem
(Theorem
3.2.1).
3.2.13
(a)
Let
A
be
a
set
and
f
:
A
→
P(A)
a
function.
By
considering
{a
∈
A
|
a
<
f
(a)},
prove
that
f
is
not
surjective.
(b)
Deduce
Cantor’s
theorem
(Theorem
3.2.2):
|A|
<
|P(A)|
for
all
sets
A.
3.2.14
(a)
Let
A
be
a
category.
Suppose
there
exists
a
functor
U
:
A
→
Set
such
that
U
has
a
left
adjoint
and
for
at
least
one
A
∈
A
,
the
set
U(A)
has
at
least
two
elements.
Prove
that
for
any
set
I
and
any
family
(A
i
)
i∈I
of
objects
of
A
,
there
is
some
object
of
A
not
isomorphic
to
A
i
for
any
i
∈
I.
(Hint:
use
Exercise
2.3.11.)
78
Interlude
on
sets
(b)
Let
A
be
a
category
satisfying
the
assumption
of
(a).
Prove
that
A
is
not
essentially
small.
(c)
Deduce
that
none
of
the
categories
Set,
Vect
k
,
Grp,
Ab,
Ring,
and
Top
is
essentially
small.
3.2.15
(a)
(b)
(c)
(d)
(e)
Which
of
the
following
categories
are
small?
Which
are
locally
small?
Mon,
the
category
of
monoids;
Z,
the
group
of
integers,
viewed
as
a
one-object
category;
Z,
the
ordered
set
of
integers;
Cat,
the
category
of
small
categories;
the
multiplicative
monoid
of
cardinals.
3.2.16
Let
O
:
Cat
→
Set
be
the
functor
sending
a
small
category
to
its
set
of
objects.
Exhibit
a
chain
of
adjoints
C
a
D
a
O
a
I.
3.3
Historical
remarks
The
set
theory
that
we
began
to
develop
in
Section
3.1
is
rather
different
from
what
many
mathematicians
think
of
as
set
theory.
Here
I
will
explain
what
the
socially
dominant
version
of
set
theory
is,
why,
despite
its
dominance,
it
is
the
object
of
widespread
suspicion,
and
why
the
kind
of
set
theory
outlined
here
is
a
more
accurate
reflection
of
how
mathematicians
use
sets
in
practice.
Cantor’s
set
theory
The
creation
of
set
theory
is
generally
credited
to
the
German
mathematician
Georg
Cantor,
in
the
late
nineteenth
century.
Previ-
ously,
sets
had
seldom
been
regarded
as
entities
worthy
of
study
in
their
own
right;
but
Cantor,
originally
motivated
by
a
problem
in
Fourier
analysis,
de-
veloped
an
extensive
theory.
Among
many
other
things,
he
showed
that
there
are
different
sizes
of
infinity,
proving,
for
instance,
that
there
is
no
bijection
between
N
and
R.
Cantor’s
theory
met
all
the
resistance
that
typically
greets
a
really
new
idea.
His
work
was
criticized
as
nonsensical,
as
meaningless,
as
far
too
abstract;
then
later,
as
all
very
well
but
of
no
use
to
the
mainstream
of
mathematics.
Kronecker,
an
important
mathematician
of
the
day,
called
him
a
charlatan
and
a
corrupter
of
youth.
But
nowadays,
the
basics
of
Cantor’s
work
are
on
nearly
every
undergraduate
mathematics
syllabus.
Times
change.
In
the
modern
style
of
mathematics,
almost
every
definition,
when
unravelled
sufficiently,
depends
on
the
notion
of
set.
But
pre-Cantor,
this
was
not
so.
It
is
interesting
to
try
to
understand
the
outlook
of
mathematicians
3.3
Historical
remarks
79
of
the
time,
who
had
successfully
developed
sophisticated
subjects
such
as
complex
analysis
and
Galois
theory
without
depending
on
this
notion
that
we
now
regard
as
fundamental.
Before
continuing
with
the
history,
we
need
to
discuss
another
fundamental
concept.
√
Types
Suppose
someone
asks
you
‘is
2
√
=
π?’
Your
answer
is,
of
course,
‘no’.
Now
suppose
someone
asks
you
‘is
2
=
log?’
You
might
frown
and
wonder
if
you
had
heard
right,
and
perhaps
your
answer
would
again
be
‘no’;
√
but
it
would
be
a
different
kind
of
‘no’.
After
all,
2
is
a
number,
whereas
log
is
a
function,
so
it
is
inconceivable
that
they
could
be
equal.
A
better
answer
would
be
‘your
question
makes
no
sense’.
This
illustrates
the
idea
of
types.
The
square
root
of
2
is
a
real
number,
Q
is
d
a
field,
S
3
is
a
group,
log
is
a
function
from
(0,
∞)
to
R,
and
dx
is
an
operation
that
takes
as
input
one
function
from
R
to
R
and
produces
as
output
another
√
such
function.
One
says
that
the
type
of
2
is
‘real
number’,
the
type
of
Q
is
‘field’,
and
so
on.
We
all
have
an
inbuilt
sense
of
type,
and
it
would
not
usually
occur
to
us
to
ask
whether
two
things
of
different
type
were
equal.
You
may
have
met
this
idea
before
if
you
have
programmed
computers.
Many
programming
languages
require
you
to
declare
the
type
of
a
variable
before
you
first
use
it.
For
example,
you
might
declare
that
x
is
to
be
a
variable
of
type
‘real
number’,
n
a
variable
of
type
‘integer’,
M
a
variable
of
type
‘3
×
3
matrix
of
lists
of
binary
digits’,
and
so
on.
The
distinction
between
different
types
of
object
has
always
been
instinc-
tively
understood.
At
the
beginning
of
the
twentieth
century,
however,
events
took
a
strange
turn.
Membership-based
set
theory
Those
who
came
after
Cantor
sought
to
com-
pile
a
definitive
list
of
assumptions
to
be
made
about
sets:
an
axiomatization
of
set
theory.
The
list
they
arrived
at,
in
the
early
years
of
the
twentieth
cen-
tury,
is
known
as
ZFC
(Zermelo–Fraenkel
with
Choice).
It
soon
became
the
standard,
and
it
is
the
only
kind
of
axiomatic
set
theory
that
most
present-day
mathematicians
know.
The
axiomatization
of
Zermelo
et
al.
was
in
some
ways
similar
to
the
one
that
we
were
working
towards
in
the
first
section
of
this
chapter.
But
there
is
at
least
one
crucial
difference:
whereas
we
took
sets
and
functions
as
our
basic
concepts,
they
took
sets
and
membership.
At
first
sight,
this
difference
might
seem
mild.
But
when
the
membership-
based
approach
is
used
as
a
foundation
on
which
to
build
the
rest
of
mathemat-
ics,
several
bizarre
features
become
apparent:
80
Interlude
on
sets
•
In
the
Zermelo
approach,
everything
is
a
set.
For
instance,
a
function
is
de-
fined
as
a
set
with
certain
properties.
Many
other
things
that
you
would
√
not
think
of
as
being
sets
are,
nevertheless,
treated
as
sets:
the
number
2
is
a
d
set,
the
function
log
is
a
set,
the
operator
dx
is
a
set,
and
so
on.
You
might
wonder
how
this
is
possible.
Perhaps
it
is
useful
to
compare
data
storage
in
a
computer,
where
files
of
all
different
types
(text,
sound,
images,
and
so
on)
are
ultimately
encoded
as
sequences
of
0s
and
1s.
To
give
an
example,
in
the
membership-based
set
theory
presented
in
most
books,
the
number
4
is
encoded
as
the
set
{∅,
{∅},
{∅,
{∅}},
{∅,
{∅},
{∅,
{∅}}}}.
•
The
virtue
of
this
approach
is
its
simplicity:
everything
is
a
set!
But
the
price
to
be
paid
is
very
high:
we
lose
the
fundamental
notion
of
type,
precisely
because
everything
is
regarded
as
being
of
type
‘set’.
•
In
the
Zermelo
approach,
the
elements
of
sets
are
always
sets
too.
This
is
in
conflict
with
ordinary
mathematics.
For
instance,
in
ordinary
mathematics,
R
is
certainly
a
set,
but
real
numbers
themselves
are
not
regarded
as
sets.
(After
all,
what
is
an
element
of
π?)
•
In
this
approach,
membership
is
a
global
relation,
meaning
that
for
any
two
sets
A
and
B,
it
makes
sense
to
ask
whether
A
∈
B.
Since
this
approach
views
everything
as
a
√
set,
it
makes
sense
to
ask
such
apparently
nonsensical
questions
as
‘is
Q
∈
2?’
Further
still,
the
axioms
of
ZFC
imply
that
we
can
form
the
intersection
A∩
B
of
any
sets
A
and
B.
(Its
elements
are
those
sets
C
for
which
C
∈
A
and
C
∈
B.)
This
makes
possible
further
nonsensical
questions
such
as
‘does
the
cyclic
group
of
order
10
have
nonempty
intersection
with
Z?’
The
answers
to
these
nonsensical
questions
depend
on
the
fine
detail
of
how
mathematical
objects
(numbers,
functions,
groups,
etc.)
are
encoded
as
sets.
Even
devotees
of
the
membership-based
approach
agree
that
this
encoding
is
a
matter
of
convention,
just
like
a
word
processor’s
encoding
of
a
document
as
a
string
of
0s
and
1s.
So
the
answers
to
these
questions
are
meaningless.
Set
theory
today
It
should
now
be
apparent
why
many
modern-day
mathe-
maticians
are
suspicious
of
set
theory.
However
often
they
are
told
that
it
is
‘the
foundation
of
mathematics’,
they
feel
that
much
of
it
is
irrelevant
to
their
concerns.
To
some
extent,
this
is
justified.
But
it
is
also
a
symptom
of
the
historical
dominance
of
membership-based
set
theory:
most
mathematicians
do
not
re-
alize
that
there
is
any
other
kind.
This
is
a
shame.
Taking
sets
and
functions
3.3
Historical
remarks
81
(rather
than
sets
and
membership)
as
the
basic
concepts
leads
to
a
theory
con-
taining
all
of
the
meaningful
results
of
Cantor
and
others,
but
with
none
of
the
aspects
that
seem
so
remote
from
the
rest
of
mathematics.
In
particular,
the
function-based
approach
respects
the
fundamental
notion
of
type.
The
function-based
approach
is,
of
course,
categorical,
and
its
advantages
are
related
to
more
general
points
about
how
mathematics
looks
through
cat-
egorical
eyes.
Objects
are
understood
through
their
place
in
the
ambient
cate-
gory.
We
get
inside
an
object
by
probing
it
with
maps
to
or
from
other
objects.
For
example,
an
element
of
a
set
A
is
a
map
1
→
A,
and
a
subset
of
A
is
a
map
A
→
2.
Probing
of
this
kind
is
the
main
theme
of
the
next
chapter.
Footnote
for
those
familiar
with
ZFC
People
brought
up
on
traditional
ax-
iomatic
set
theory
often
have
the
following
concern
when
they
come
across
categorical
set
theory
for
the
first
time.
The
objects
and
maps
of
a
category
form
a
collection
of
some
kind,
perhaps
a
set,
so
the
notion
of
category
ap-
pears
to
depend
on
some
prior
set-like
notion.
How,
then,
can
sets
be
axioma-
tized
categorically?
Is
that
not
circular?
It
is
not,
because
sets
can
be
axiomatized
categorically
without
mentioning
categories
once.
To
see
how,
let
us
first
recall
the
shape
of
the
ZFC
axiomati-
zation
of
sets.
Informally,
it
looks
like
this:
•
there
are
some
things
called
sets;
•
there
is
a
binary
relation
on
sets,
called
membership
(∈);
•
some
axioms
hold.
A
categorical
axiomatization
of
sets
looks,
informally,
like
this:
•
there
are
some
things
called
sets;
•
for
each
set
A
and
set
B,
there
are
some
things
called
functions
from
A
to
B;
•
to
each
function
f
from
A
to
B
and
function
g
from
B
to
C,
there
is
assigned
a
function
g
◦
f
from
A
to
C;
•
some
axioms
hold.
Making
precise
such
phrases
as
‘some
things’
requires
delicacy,
as
will
be
fa-
miliar
to
anyone
who
has
done
a
logic
course.
But
the
difficulties
are
no
worse
for
categorical
axiomatizations
of
sets
than
for
membership-based
axiomatiza-
tions
such
as
ZFC.
One
popular
choice
of
categorical
axioms
for
set
theory
can
be
summarized
informally
as
follows.
82
Interlude
on
sets
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Composition
of
functions
is
associative
and
has
identities.
There
is
a
terminal
set.
There
is
a
set
with
no
elements.
A
function
is
determined
by
its
effect
on
elements.
Given
sets
A
and
B,
one
can
form
their
product
A
×
B.
Given
sets
A
and
B,
one
can
form
the
set
of
functions
from
A
to
B.
Given
f
:
A
→
B
and
b
∈
B,
one
can
form
the
inverse
image
f
−1
{b}.
The
subsets
of
a
set
A
correspond
to
the
functions
from
A
to
{0,
1}.
The
natural
numbers
form
a
set.
Every
surjection
has
a
section.
This
informal
summary
uses
terms
such
as
‘element’
and
‘inverse
image’,
which
can
be
defined
in
terms
of
the
basic
concepts
of
set,
function
and
compo-
sition.
For
instance,
an
element
of
a
set
A
is
defined
as
a
map
from
the
terminal
set
to
A.
It
is
certainly
convenient
to
express
these
axioms
in
terms
of
categories.
For
example,
the
first
axiom
says
that
sets
and
functions
form
a
category,
and
all
ten
together
can
be
expressed
in
categorical
jargon
as
‘sets
and
functions
form
a
well-pointed
topos
with
natural
numbers
object
and
choice’.
But
in
order
to
state
the
axioms,
it
is
not
necessary
to
appeal
to
any
general
notion
of
category.
They
can
be
expressed
directly
in
terms
of
sets
and
functions.
For
details,
see
Lawvere
and
Rosebrugh
(2003)
or
Leinster
(2014).
Exercise
3.3.1
Choose
a
mathematician
at
random.
Ask
them
whether
they
can
ac-
curately
state
any
axiomatization
of
sets
(without
looking
it
up).
If
not,
ask
them
what
operating
principles
they
actually
use
when
handling
sets
in
their
day-to-day
work.
4
Representables
A
category
is
a
world
of
objects,
all
looking
at
one
another.
Each
sees
the
world
from
a
different
viewpoint.
Consider,
for
instance,
the
category
of
topological
spaces,
and
let
us
ask
how
it
looks
when
viewed
from
the
one-point
space
1.
A
map
from
1
to
a
space
X
is
essentially
the
same
thing
as
a
point
of
X,
so
we
might
say
that
1
‘sees
points’.
Similarly,
a
map
from
R
to
a
space
X
could
reasonably
be
called
a
curve
in
X,
and
in
this
sense,
R
sees
curves.
Now
consider
the
category
of
groups.
A
map
from
the
infinite
cyclic
group
Z
to
a
group
G
amounts
to
an
element
of
G.
(For
given
g
∈
G,
there
is
a
unique
homomorphism
φ
:
Z
→
G
such
that
φ(1)
=
g.)
So,
Z
sees
elements.
Similarly,
if
p
is
a
prime
number
then
the
cyclic
group
Z/pZ
sees
elements
of
order
1
or
p.
Any
ring
homomorphism
between
fields
is
injective,
so
in
the
category
of
fields,
a
map
K
→
L
is
a
way
of
realizing
L
as
an
extension
of
K.
Hence
each
field
K
sees
the
extensions
of
itself.
If
K
and
L
are
fields
of
different
charac-
teristic
then
there
are
no
homomorphisms
between
K
and
L,
so
the
category
of
fields
is
the
union
of
disjoint
subcategories
Field
0
,
Field
2
,
Field
3
,
Field
5
,
.
.
.
consisting
of
the
fields
of
characteristics
0,
2,
3,
5,
.
.
.
.
Each
field
is
blind
to
the
fields
of
different
characteristic.
In
the
ordered
set
(R,
≤),
the
object
0
sees
whether
a
number
is
nonnegative.
In
other
words,
if
x
is
nonnegative
then
there
is
one
map
0
→
x,
and
if
not,
there
are
none.
We
can
also
ask
the
dual
question:
fixing
an
object
of
a
category,
what
are
the
maps
into
it?
Let
S
be
the
two-element
set,
for
instance.
For
an
arbitrary
set
X,
the
maps
from
X
to
S
correspond
to
the
subsets
of
X
(as
we
saw
in
Section
3.1).
Now
give
S
the
topology
in
which
one
of
the
singleton
subsets
is
open
but
the
other
is
not.
For
any
topological
space
X,
the
continuous
maps
from
X
into
S
correspond
to
the
open
subsets
of
X.
83
84
Representables
This
chapter
explores
the
theme
of
how
each
object
sees
and
is
seen
by
the
category
in
which
it
lives.
We
are
naturally
led
to
the
notion
of
representable
functor,
which
(after
adjunctions)
provides
our
second
approach
to
the
idea
of
universal
property.
4.1
Definitions
and
examples
Fix
an
object
A
of
a
category
A
.
We
will
consider
the
totality
of
maps
out
of
A.
To
each
B
∈
A
,
there
is
assigned
the
set
(or
class)
A
(A,
B)
of
maps
from
A
to
B.
The
content
of
the
following
definition
is
that
this
assignation
is
functorial
in
B:
any
map
B
→
B
0
induces
a
function
A
(A,
B)
→
A
(A,
B
0
).
Definition
4.1.1
functor
Let
A
be
a
locally
small
category
and
A
∈
A
.
We
define
a
H
A
=
A
(A,
−)
:
A
→
Set
as
follows:
•
for
objects
B
∈
A
,
put
H
A
(B)
=
A
(A,
B);
g
•
for
maps
B
−→
B
0
in
A
,
define
H
A
(g)
=
A
(A,
g)
:
A
(A,
B)
→
A
(A,
B
0
)
by
p
7→
g
◦
p
for
all
p
:
A
→
B.
Remarks
4.1.2
(a)
Recall
that
‘locally
small’
means
that
each
class
A
(A,
B)
is
in
fact
a
set.
This
hypothesis
is
clearly
necessary
in
order
for
the
defini-
tion
to
make
sense.
(b)
Sometimes
H
A
(g)
is
written
as
g
◦
−
or
g
∗
.
All
three
forms,
as
well
as
A
(A,
g),
are
in
use.
Definition
4.1.3
Let
A
be
a
locally
small
category.
A
functor
X
:
A
→
Set
is
representable
if
X
H
A
for
some
A
∈
A
.
A
representation
of
X
is
a
choice
of
an
object
A
∈
A
and
an
isomorphism
between
H
A
and
X.
Representable
functors
are
sometimes
just
called
‘representables’.
Only
set-
valued
functors
(that
is,
functors
with
codomain
Set)
can
be
representable.
4.1
Definitions
and
examples
85
Example
4.1.4
Consider
H
1
:
Set
→
Set,
where
1
is
the
one-element
set.
Since
a
map
from
1
to
a
set
B
amounts
to
an
element
of
B,
we
have
H
1
(B)
B
for
each
B
∈
Set.
It
is
easily
verified
that
this
isomorphism
is
natural
in
B,
so
H
1
is
isomorphic
to
the
identity
functor
1
Set
.
Hence
1
Set
is
representable.
Example
4.1.5
All
of
the
‘seeing’
functors
in
the
introduction
to
this
chapter
are
representable.
The
forgetful
functor
Top
→
Set
is
isomorphic
to
H
1
=
Top(1,
−),
and
the
forgetful
functor
Grp
→
Set
is
isomorphic
to
Grp(Z,
−).
For
each
prime
p,
there
is
a
functor
U
p
:
Grp
→
Set
defined
on
objects
by
U
p
(G)
=
{elements
of
G
of
order
1
or
p},
and
as
claimed
above,
U
p
Grp(Z/pZ,
−)
(Exercise
4.1.28).
Hence
U
p
is
representable.
Example
4.1.6
There
is
a
functor
ob
:
Cat
→
Set
sending
a
small
category
to
its
set
of
objects.
(The
category
Cat
was
introduced
in
Definition
3.2.10.)
It
is
representable.
Indeed,
consider
the
terminal
category
1
(with
one
object
and
only
the
identity
map).
A
functor
from
1
to
a
category
B
simply
picks
out
an
object
of
B.
Thus,
H
1
(B)
ob
B.
Again,
it
is
easily
verified
that
this
isomorphism
is
natural
in
B;
hence
ob
Cat(1,
−).
It
can
be
shown
similarly
that
the
functor
Cat
→
Set
sending
a
small
category
to
its
set
of
maps
is
representable
(Exercise
4.1.31).
Example
4.1.7
Let
M
be
a
monoid,
regarded
as
a
one-object
category.
Recall
from
Example
1.2.8
that
a
set-valued
functor
on
M
is
just
an
M-set.
Since
the
category
M
has
only
one
object,
there
is
only
one
representable
functor
on
it
(up
to
isomorphism).
As
an
M-set,
the
unique
representable
is
the
so-called
left
regular
representation
of
M,
that
is,
the
underlying
set
of
M
acted
on
by
multiplication
on
the
left.
Example
4.1.8
Let
Toph
∗
be
the
category
whose
objects
are
topological
spa-
ces
equipped
with
a
basepoint
and
whose
arrows
are
homotopy
classes
of
basepoint-preserving
continuous
maps.
Let
S
1
∈
Toph
∗
be
the
circle.
Then
for
any
object
X
∈
Toph
∗
,
the
maps
S
1
→
X
in
Toph
∗
are
the
elements
of
the
fundamental
group
π
1
(X).
Formally,
this
says
that
the
composite
functor
π
1
U
Toph
∗
−→
Grp
−→
Set
is
isomorphic
to
Toph
∗
(S
1
,
−).
In
particular,
it
is
representable.
86
Representables
Example
4.1.9
Fix
a
field
k
and
vector
spaces
U
and
V
over
k.
There
is
a
functor
Bilin(U,
V;
−)
:
Vect
k
→
Set
whose
value
Bilin(U,
V;
W)
at
W
∈
Vect
k
is
the
set
of
bilinear
maps
U
×
V
→
W.
It
can
be
shown
that
this
functor
is
representable;
in
other
words,
there
is
a
space
T
with
the
property
that
Bilin(U,
V;
W)
Vect
k
(T,
W)
naturally
in
W.
This
T
is
the
tensor
product
U
⊗
V,
which
we
met
just
after
the
proof
of
Lemma
0.7.
Adjunctions
give
rise
to
representable
functors
in
the
following
way.
Lemma
4.1.10
Let
A
o
/
F
⊥
G
B
be
locally
small
categories,
and
let
A
∈
A
.
Then
the
functor
A
(A,
G(−))
:
B
→
Set
H
A
G
(that
is,
the
composite
B
−→
A
−→
Set)
is
representable.
Proof
We
have
A
(A,
G(B))
B(F(A),
B)
for
each
B
∈
B.
If
we
can
show
that
this
isomorphism
is
natural
in
B,
then
we
will
have
proved
that
A
(A,
G(−))
is
isomorphic
to
H
F(A)
and
is
therefore
q
representable.
So,
let
B
−→
B
0
be
a
map
in
B.
We
must
show
that
the
square
A
(A,
G(B))
G(q)◦−
A
(A,
G(B
0
))
/
B(F(A),
B)
q◦−
/
B(F(A),
B
0
)
commutes,
where
the
horizontal
arrows
are
the
bijections
provided
by
the
ad-
junction.
For
f
:
A
→
G(B),
we
have
f
_
G(q)
◦
f
/
f
¯
_
q
◦
f
¯
/
G(q)
◦
f
,
so
we
must
prove
that
q
◦
f
¯
=
G(q)
◦
f
.
This
follows
immediately
from
the
naturality
condition
(2.2)
in
the
definition
of
adjunction
(with
g
=
f
¯
).
4.1
Definitions
and
examples
87
You
would
not
expect
a
randomly-chosen
functor
into
Set
to
be
represen-
table.
In
some
sense,
rather
few
functors
are.
However,
forgetful
functors
do
tend
to
be
representable:
Proposition
4.1.11
Any
set-valued
functor
with
a
left
adjoint
is
representable.
Proof
Let
G
:
A
→
Set
be
a
functor
with
a
left
adjoint
F.
Write
1
for
the
one-point
set.
Then
G(A)
Set(1,
G(A))
naturally
in
A
∈
A
(by
Example
4.1.4),
that
is,
G
Set(1,
G(−)).
So
by
Lemma
4.1.10,
G
is
representable;
indeed,
G
H
F(1)
.
Example
4.1.12
Several
of
the
examples
of
representables
mentioned
above
arise
as
in
Proposition
4.1.11.
For
instance,
U
:
Top
→
Set
has
a
left
adjoint
D
(Example
2.1.5),
and
D(1)
1,
so
we
recover
the
result
that
U
H
1
.
Similarly,
Exercise
3.2.16
asked
you
to
construct
a
left
adjoint
D
to
the
objects
functor
ob
:
Cat
→
Set.
This
functor
D
satisfies
D(1)
1,
proving
again
that
ob
H
1
.
Example
4.1.13
The
forgetful
functor
U
:
Vect
k
→
Set
is
representable,
since
it
has
a
left
adjoint.
Indeed,
if
F
denotes
the
left
adjoint
then
F(1)
is
the
1-dimensional
vector
space
k,
so
U
H
k
.
This
is
also
easy
to
see
directly:
a
map
from
k
to
a
vector
space
V
is
uniquely
determined
by
the
image
of
1,
which
can
be
any
element
of
V;
hence
Vect
k
(k,
V)
U(V)
naturally
in
V.
Example
4.1.14
Examples
2.1.3
began
with
the
declaration
that
forgetful
functors
between
categories
of
algebraic
structures
usually
have
left
adjoints.
Take
the
category
CRing
of
commutative
rings
and
the
forgetful
functor
U
:
CRing
→
Set.
This
general
principle
suggests
that
U
has
a
left
adjoint,
and
Proposition
4.1.11
then
tells
us
that
U
is
representable.
Let
us
see
how
this
works
explicitly.
Given
a
set
S
,
let
Z[S
]
be
the
ring
of
polynomials
over
Z
in
commuting
variables
x
s
(s
∈
S
).
(This
was
called
F(S
)
in
Example
1.2.4(b).)
Then
S
7→
Z[S
]
defines
a
functor
Set
→
CRing,
and
this
is
left
adjoint
to
U.
Hence
U
H
Z[x]
.
Again,
this
can
be
verified
directly:
for
any
ring
R,
the
maps
Z[x]
→
R
correspond
one-to-one
with
the
elements
of
R
(Exercises
0.13
and
4.1.29).
We
have
defined,
for
each
object
A
of
our
category
A
,
a
functor
H
A
∈
[A
,
Set].
This
describes
how
A
sees
the
world.
As
A
varies,
the
view
varies.
On
the
other
hand,
it
is
always
the
same
world
being
seen,
so
the
different
views
from
different
objects
are
somehow
related.
(Compare
aerial
photos
taken
from
a
moving
aeroplane,
which
agree
well
enough
on
their
overlaps
that
they
can
be
88
Representables
patched
together
to
make
one
big
picture.)
So
the
family
H
A
A∈A
of
‘views’
has
some
consistency
to
it.
What
this
means
is
that
whenever
there
is
a
map
0
between
objects
A
and
A
0
,
there
is
also
a
map
between
H
A
and
H
A
.
f
Precisely,
a
map
A
0
−→
A
induces
a
natural
transformation
H
A
A
H
(
H
f
6
Set,
A
0
whose
B-component
(for
B
∈
A
)
is
the
function
H
A
(B)
=
A
(A,
B)
→
p
7→
0
H
A
(B)
=
A
(A
0
,
B)
p
◦
f.
Again,
H
f
goes
by
a
variety
of
other
names:
A
(
f,
−),
f
∗
,
and
−
◦
f
.
Note
the
reversal
of
direction!
Each
functor
H
A
is
covariant,
but
they
come
together
to
form
a
contravariant
functor,
as
in
the
following
definition.
Definition
4.1.15
Let
A
be
a
locally
small
category.
The
functor
H
•
:
A
op
→
[A
,
Set]
is
defined
on
objects
A
by
H
•
(A)
=
H
A
and
on
maps
f
by
H
•
(
f
)
=
H
f
.
The
symbol
•
is
another
type
of
blank,
like
−.
All
of
the
definitions
presented
so
far
in
this
chapter
can
be
dualized.
At
the
formal
level,
this
is
trivial:
reverse
all
the
arrows,
so
that
every
A
becomes
an
A
op
and
vice
versa.
But
in
our
usual
examples,
the
flavour
is
different.
We
are
no
longer
asking
what
objects
see,
but
how
they
are
seen.
Let
us
first
dualize
Definition
4.1.1.
Definition
4.1.16
a
functor
Let
A
be
a
locally
small
category
and
A
∈
A
.
We
define
H
A
=
A
(−,
A)
:
A
op
→
Set
as
follows:
•
for
objects
B
∈
A
,
put
H
A
(B)
=
A
(B,
A);
g
•
for
maps
B
0
−→
B
in
A
,
define
H
A
(g)
=
A
(g,
A)
=
g
∗
=
−
◦
g
:
A
(B,
A)
→
A
(B
0
,
A)
by
p
7→
p
◦
g
for
all
p
:
B
→
A.
4.1
Definitions
and
examples
89
If
you
know
about
dual
vector
spaces,
this
construction
will
seem
familiar.
In
particular,
you
will
not
be
surprised
that
a
map
B
0
→
B
induces
a
map
in
the
opposite
direction,
H
A
(B)
→
H
A
(B
0
).
We
now
define
representability
for
contravariant
set-valued
functors.
Stri-
ctly
speaking,
this
is
unnecessary,
as
a
contravariant
functor
on
A
is
a
covariant
functor
on
A
op
,
and
we
already
know
what
it
means
for
a
covariant
set-valued
functor
to
be
representable.
But
it
is
useful
to
have
a
direct
definition.
Definition
4.1.17
Let
A
be
a
locally
small
category.
A
functor
X
:
A
op
→
Set
is
representable
if
X
H
A
for
some
A
∈
A
.
A
representation
of
X
is
a
choice
of
an
object
A
∈
A
and
an
isomorphism
between
H
A
and
X.
Example
4.1.18
There
is
a
functor
P
:
Set
op
→
Set
sending
each
set
B
to
its
power
set
P(B),
and
defined
on
maps
g
:
B
0
→
B
by
(P(g))(U)
=
g
−1
U
for
all
U
∈
P(B).
(Here
g
−1
U
denotes
the
inverse
image
or
preimage
of
U
under
g,
defined
by
g
−1
U
=
{x
0
∈
B
0
|
g(x
0
)
∈
U}.)
As
we
saw
in
Section
3.1,
a
subset
amounts
to
a
map
into
the
two-point
set
2.
Precisely
put,
P
H
2
.
Example
4.1.19
Similarly,
there
is
a
functor
O
:
Top
op
→
Set
defined
on
objects
B
by
taking
O(B)
to
be
the
set
of
open
subsets
of
B.
If
S
denotes
the
two-point
topological
space
in
which
exactly
one
of
the
two
single-
ton
subsets
is
open,
then
continuous
maps
from
a
space
B
into
S
correspond
naturally
to
open
subsets
of
B
(Exercise
4.1.30).
Hence
O
H
S
,
and
O
is
representable.
Example
4.1.20
In
Example
1.2.11,
we
defined
a
functor
C
:
Top
op
→
Ring,
assigning
to
each
space
the
ring
of
continuous
real-valued
functions
on
it.
The
composite
functor
C
U
Top
op
−→
Ring
−→
Set
is
representable,
since
by
definition,
U(C(X))
=
Top(X,
R)
for
topological
spaces
X.
Previously,
we
assembled
the
covariant
representables
H
A
A∈A
into
one
big
functor
H
•
.
We
now
do
the
same
for
the
contravariant
representables
H
A
A∈A
.
90
Representables
f
Any
map
A
−→
A
0
in
A
induces
a
natural
transformation
H
A
A
op
H
f
'
7
Set
H
A
0
(also
called
A
(−,
f
),
f
∗
or
f
◦
−),
whose
component
at
an
object
B
∈
A
is
H
A
(B)
=
A
(B,
A)
→
p
7→
H
A
0
(B)
=
A
(B,
A
0
)
f
◦
p.
Definition
4.1.21
Let
A
be
a
locally
small
category.
The
Yoneda
embed-
ding
of
A
is
the
functor
H
•
:
A
→
[A
op
,
Set]
defined
on
objects
A
by
H
•
(A)
=
H
A
and
on
maps
f
by
H
•
(
f
)
=
H
f
.
Here
is
a
summary
of
the
definitions
so
far.
H
A
For
each
A
∈
A
,
we
have
a
functor
A
−→
Set.
Putting
them
all
together
gives
a
functor
A
op
−→
[A
,
Set].
For
each
A
∈
A
,
we
have
a
functor
A
op
−→
Set.
Putting
them
all
together
gives
a
functor
A
−→
[A
op
,
Set].
H
•
H
A
H
•
The
second
pair
of
functors
is
the
dual
of
the
first.
Both
involve
contravariance;
it
cannot
be
avoided.
In
the
theory
of
representable
functors,
it
does
not
make
much
difference
whether
we
work
with
the
first
or
the
second
pair.
Any
theorem
that
we
prove
about
one
dualizes
to
give
a
theorem
about
the
other.
We
choose
to
work
with
the
second
pair,
the
H
A
s
and
H
•
.
In
a
sense
to
be
explained,
H
•
‘embeds’
A
into
[A
op
,
Set].
This
can
be
useful,
because
the
category
[A
op
,
Set]
has
some
good
properties
that
A
might
not
have.
Exercise
4.1.27
asks
you
to
prove
that
H
•
is
injective
on
isomorphism
classes
of
objects.
It
is
strongly
recommended
that
you
do
it
before
reading
on,
as
it
encapsulates
the
key
ideas
of
the
rest
of
this
chapter.
There
is
one
more
functor
to
define.
It
unifies
the
first
and
second
pairs
of
functors
shown
above.
Definition
4.1.22
Let
A
be
a
locally
small
category.
The
functor
Hom
A
:
A
op
×
A
→
Set
91
4.1
Definitions
and
examples
is
defined
by
(A,
O
B)
7→
g
7→
f
(A
0
,
B
0
)
A
(A,
B)
g◦−◦
f
A
(A
0
,
B
0
).
7→
In
other
words,
Hom
A
(A,
B)
=
A
(A,
B)
and
(Hom
A
(
f,
g))(p)
=
g
◦
p
◦
f
,
f
p
g
whenever
A
0
−→
A
−→
B
−→
B
0
.
Remarks
4.1.23
(a)
The
existence
of
the
functor
Hom
A
is
something
like
the
fact
that
for
a
metric
space
(X,
d),
the
metric
is
itself
a
continuous
map
d
:
X
×
X
→
R.
(If
we
take
two
points
and
move
each
one
slightly,
the
distance
between
them
changes
only
slightly.)
(b)
In
terms
of
Exercise
1.2.25,
Hom
A
is
the
functor
A
op
×
A
→
Set
corre-
sponding
to
the
families
of
functors
H
A
A∈A
and
H
B
B∈A
.
(c)
In
Example
2.1.6,
we
saw
that
for
any
set
B,
there
is
an
adjunction
(−×B)
a
(−)
B
of
functors
Set
→
Set.
Similarly,
for
any
category
B,
there
is
an
adjunction
(−
×
B)
a
[B,
−]
of
functors
CAT
→
CAT;
in
other
words,
there
is
a
canonical
bijection
CAT(A
×
B,
C
)
CAT(A
,
[B,
C
])
for
A
,
B,
C
∈
CAT.
Under
this
bijection,
the
functors
Hom
A
:
A
op
×
A
→
Set,
H
•
:
A
op
→
[A
,
Set]
correspond
to
one
another.
Thus,
Hom
A
carries
the
same
information
as
H
•
(or
H
•
),
presented
slightly
differently.
Remark
4.1.24
We
can
now
explain
the
naturality
in
the
definition
of
adjunc-
F
/
B
.
They
give
tion
(Definition
2.1.1).
Take
categories
and
functors
A
o
G
rise
to
functors
A
op
×
B
F
op
×1
1×G
/
A
op
×
A
Hom
A
B
op
×
B
Hom
B
/
Set.
The
composite
functor
↓→
sends
(A,
B)
to
B(F(A),
B);
it
can
be
written
as
B(F(−),
−).
The
composite
→↓
sends
(A,
B)
to
A
(A,
G(B)).
Exercise
4.1.32
asks
you
to
show
that
these
two
functors
B(F(−),
−),
A
(−,
G(−))
:
A
op
×
B
→
Set
92
Representables
are
naturally
isomorphic
if
and
only
if
F
and
G
are
adjoint.
This
justifies
the
claim
in
Remark
2.1.2(a):
the
naturality
requirements
(2.2)
and
(2.3)
in
the
definition
of
adjunction
simply
assert
that
two
particular
functors
are
naturally
isomorphic.
Objects
of
an
arbitrary
category
do
not
have
elements
in
any
obvious
sense.
However,
sets
certainly
have
elements,
and
we
have
observed
that
an
element
of
a
set
A
is
the
same
thing
as
a
map
1
→
A.
This
inspires
the
following
definition.
Definition
4.1.25
Let
A
be
an
object
of
a
category.
A
generalized
element
of
A
is
a
map
with
codomain
A.
A
map
S
→
A
is
a
generalized
element
of
A
of
shape
S
.
‘Generalized
element’
is
nothing
more
than
a
synonym
of
‘map’,
but
some-
times
it
is
useful
to
think
of
maps
as
generalized
elements.
For
example,
when
A
is
a
set,
a
generalized
element
of
A
of
shape
1
is
an
ordinary
element
of
A,
and
a
generalized
element
of
A
of
shape
N
is
a
sequence
in
A.
In
the
category
of
topological
spaces,
the
generalized
elements
of
shape
1
(the
one-point
space)
are
the
points,
and
the
generalized
elements
of
shape
S
1
(the
circle)
are,
by
definition,
loops.
As
this
suggests,
in
categories
of
geometric
objects,
we
might
equally
well
say
‘figures
of
shape
S
’.
In
algebra,
we
are
often
interested
in
solutions
to
equations
such
as
x
2
+
y
2
=
1.
Perhaps
we
begin
by
being
particularly
interested
in
solutions
in
Q,
but
then
realize
that
in
order
to
study
rational
solutions,
it
will
be
helpful
to
study
solutions
in
other
rings
first.
(This
is
often
a
fruitful
strategy.)
Given
a
ring
A,
a
pair
(a,
b)
∈
A
×
A
satisfying
a
2
+
b
2
=
1
amounts
to
a
homomorphism
of
rings
Z[x,
y]/(x
2
+
y
2
−
1)
→
A.
Thus,
the
solutions
to
our
equation
(in
any
ring)
can
be
seen
as
the
generalized
elements
of
shape
Z[x,
y]/(x
2
+
y
2
−
1).
For
an
object
S
of
a
category
A
,
the
functor
H
S
:
A
→
Set
sends
an
object
to
its
set
of
generalized
elements
of
shape
S
.
The
functoriality
tells
us
that
any
map
A
→
B
in
A
transforms
S
-elements
of
A
into
S
-elements
of
B.
For
example,
taking
A
=
Top
and
S
=
S
1
,
any
continuous
map
A
→
B
transforms
loops
in
A
into
loops
in
B.
Exercises
4.1.26
Find
three
examples
of
representable
functors
not
mentioned
above.
93
4.2
The
Yoneda
lemma
4.1.27
Let
A
be
a
locally
small
category,
and
let
A,
A
0
∈
A
with
H
A
H
A
0
.
Prove
directly
that
A
A
0
.
4.1.28
Let
p
be
a
prime
number.
Show
that
the
functor
U
p
:
Grp
→
Set
defined
in
Example
4.1.5
is
isomorphic
to
Grp(Z/pZ,
−).
(To
check
that
there
is
an
isomorphism
of
functors
–
that
is,
a
natural
isomorphism
–
you
will
first
need
to
define
U
p
on
maps.
There
is
only
one
sensible
way
to
do
this.)
4.1.29
Using
the
result
of
Exercise
0.13(a),
prove
that
the
forgetful
functor
CRing
→
Set
is
isomorphic
to
CRing(Z[x],
−),
as
in
Example
4.1.14.
4.1.30
The
Sierpiński
space
is
the
two-point
topological
space
S
in
which
one
of
the
singleton
subsets
is
open
but
the
other
is
not.
Prove
that
for
any
topological
space
X,
there
is
a
canonical
bijection
between
the
open
subsets
of
X
and
the
continuous
maps
X
→
S
.
Use
this
to
show
that
the
functor
O
:
Top
op
→
Set
of
Example
4.1.19
is
represented
by
S
.
4.1.31
Let
M
:
Cat
→
Set
be
the
functor
that
sends
a
small
category
A
to
the
set
of
all
maps
in
A
.
Prove
that
M
is
representable.
4.1.32
Take
locally
small
categories
A
and
B,
and
functors
A
o
F
G
/
B
.
Show
that
F
is
left
adjoint
to
G
if
and
only
if
the
two
functors
B(F(−),
−),
A
(−,
G(−))
:
A
op
×
B
→
Set
of
Remark
4.1.24
are
naturally
isomorphic.
(Hint:
this
is
made
easier
by
using
either
Exercise
1.3.29
or
Exercise
2.1.14.)
4.2
The
Yoneda
lemma
What
do
representables
see?
Recall
from
Definition
1.2.15
that
functors
A
op
→
Set
are
sometimes
called
‘presheaves’
on
A
.
So
for
each
A
∈
A
we
have
a
representable
presheaf
H
A
,
and
we
are
asking
how
the
rest
of
the
presheaf
category
[A
op
,
Set]
looks
from
the
viewpoint
of
H
A
.
In
other
words,
if
X
is
another
presheaf,
what
are
the
maps
H
A
→
X?
Newcomers
to
category
theory
commonly
find
that
the
material
presented
in
this
section
is
where
they
first
get
stuck.
Typically,
the
core
of
the
difficulty
is
in
understanding
the
question
just
asked.
Let
us
ask
it
again.
We
start
by
fixing
a
locally
small
category
A
.
We
then
take
an
object
A
∈
A
and
a
functor
X
:
A
op
→
Set.
The
object
A
gives
rise
to
another
functor
94
Representables
H
A
=
A
(−,
A)
:
A
op
→
Set.
The
question
is:
what
are
the
maps
H
A
→
X?
Since
H
A
and
X
are
both
objects
of
the
presheaf
category
[A
op
,
Set],
the
‘maps’
concerned
are
maps
in
[A
op
,
Set].
So,
we
are
asking
what
natural
trans-
formations
H
A
A
op
(
6
Set
(4.1)
X
there
are.
The
set
of
such
natural
transformations
is
called
[A
op
,
Set](H
A
,
X).
(This
is
a
special
case
of
the
notation
B(B,
B
0
)
for
the
set
of
maps
B
→
B
0
in
a
category
B.
Here,
B
=
[A
op
,
Set],
B
=
H
A
,
and
B
0
=
X.)
We
want
to
know
what
this
set
is.
There
is
an
informal
principle
of
general
category
theory
that
allows
us
to
guess
the
answer.
Look
back
at
Remarks
1.1.2(b),
1.2.2(a)
and
1.3.2(a)
on
the
definitions
of
category,
functor
and
natural
transformation.
Each
remark
is
of
the
form
‘from
input
of
one
type,
it
is
possible
to
construct
exactly
one
output
of
another
type’.
For
example,
in
Remark
1.1.2(b),
the
input
is
a
sequence
of
f
1
f
n
maps
A
0
−→
·
·
·
−→
A
n
,
the
output
is
a
map
A
0
→
A
n
,
and
the
statement
is
that
no
matter
what
we
do
with
the
input
data
f
1
,
.
.
.
,
f
n
,
there
is
only
one
map
A
0
→
A
n
that
we
can
construct.
Let
us
apply
this
principle
to
our
question.
We
have
just
seen
how,
given
as
input
an
object
A
∈
A
and
a
presheaf
X
on
A
,
we
can
construct
a
set,
namely,
[A
op
,
Set](H
A
,
X).
Are
there
any
other
ways
to
construct
a
set
from
the
same
input
data
(A,
X)?
Yes:
simply
take
the
set
X(A)!
The
informal
principle
suggests
that
these
two
sets
are
the
same:
[A
op
,
Set](H
A
,
X)
X(A)
(4.2)
for
all
A
∈
A
and
X
∈
[A
op
,
Set].
This
turns
out
to
be
true;
and
that
is
the
Yoneda
lemma.
Informally,
then,
the
Yoneda
lemma
says
that
for
any
A
∈
A
and
presheaf
X
on
A
:
A
natural
transformation
H
A
→
X
is
an
element
of
X(A).
Here
is
the
formal
statement.
The
proof
follows
shortly.
Theorem
4.2.1
(Yoneda)
Let
A
be
a
locally
small
category.
Then
[A
op
,
Set](H
A
,
X)
X(A)
naturally
in
A
∈
A
and
X
∈
[A
op
,
Set].
(4.3)
95
4.2
The
Yoneda
lemma
This
is
exactly
what
was
stated
in
(4.2),
except
that
the
word
‘naturally’
has
appeared.
Recall
from
Definition
1.3.12
that
for
functors
F,
G
:
C
→
D,
the
phrase
‘F(C)
G(C)
naturally
in
C’
means
that
there
is
a
natural
isomorphism
F
G.
So
the
use
of
this
phrase
in
the
Yoneda
lemma
suggests
that
each
side
of
(4.3)
is
functorial
in
both
A
and
X.
This
means,
for
instance,
that
a
map
X
→
X
0
must
induce
a
map
[A
op
,
Set](H
A
,
X)
→
[A
op
,
Set](H
A
,
X
0
),
and
that
not
only
does
the
isomorphism
(4.3)
hold
for
every
A
and
X,
but
also,
the
isomorphisms
can
be
chosen
in
a
way
that
is
compatible
with
these
induced
maps.
Precisely,
the
Yoneda
lemma
states
that
the
composite
functor
op
Hom
H
•
×1
[A
,Set]
/
Set
/
[A
op
,
Set]
op
×
[A
op
,
Set]
A
op
×
[A
op
,
Set]
op
7−→
7−→
[A
,
Set](H
A
,
X)
(A,
X)
(H
A
,
X)
op
is
naturally
isomorphic
to
the
evaluation
functor
A
op
×
[A
op
,
Set]
(A,
X)
→
7→
Set
X(A).
If
the
Yoneda
lemma
were
false
then
the
world
would
look
much
more
com-
plex.
For
take
a
presheaf
X
:
A
op
→
Set,
and
define
a
new
presheaf
X
0
by
X
0
=
[A
op
,
Set](H
•
,
X)
:
A
op
→
Set,
that
is,
X
0
(A)
=
[A
op
,
Set](H
A
,
X)
for
all
A
∈
A
.
Yoneda
tells
us
that
X
0
(A)
X(A)
naturally
in
A;
in
other
words,
X
0
X.
If
Yoneda
were
false
then
starting
from
a
single
presheaf
X,
we
could
build
an
infinite
sequence
X,
X
0
,
X
00
,
.
.
.
of
new
presheaves,
potentially
all
different.
But
in
reality,
the
situation
is
very
simple:
they
are
all
the
same.
The
proof
of
the
Yoneda
lemma
is
the
longest
proof
so
far.
Nevertheless,
there
is
essentially
only
one
way
to
proceed
at
each
stage.
If
you
suspect
that
you
are
one
of
those
newcomers
to
category
theory
for
whom
the
Yoneda
lemma
presents
the
first
serious
challenge,
an
excellent
exercise
is
to
work
out
the
proof
before
reading
it.
No
ingenuity
is
required,
only
an
understanding
of
all
the
terms
in
the
statement.
Proof
of
the
Yoneda
lemma
We
have
to
define,
for
each
A
and
X,
a
bijection
between
the
sets
[A
op
,
Set](H
A
,
X)
and
X(A).
We
then
have
to
show
that
our
bijection
is
natural
in
A
and
X.
96
Representables
First,
fix
A
∈
A
and
X
∈
[A
op
,
Set].
We
define
functions
[A
op
,
Set](H
A
,
X)
o
(
ˆ
)
/
X(A)
(4.4)
(
˜
)
and
show
that
they
are
mutually
inverse.
So
we
have
to
do
four
things:
define
the
function
(
ˆ
),
define
the
function
(
˜
),
show
that
(
ˆ˜
)
is
the
identity,
and
show
that
(
˜ˆ
)
is
the
identity.
•
Given
α
:
H
A
→
X,
define
α̂
∈
X(A)
by
α̂
=
α
A
(1
A
).
(How
else
could
we
possibly
define
it?)
•
Let
x
∈
X(A).
We
have
to
define
a
natural
transformation
x̃
:
H
A
→
X.
That
is,
we
have
to
define
for
each
B
∈
A
a
function
x̃
B
:
H
A
(B)
=
A
(B,
A)
→
X(B)
and
show
that
the
family
x̃
=
(
x̃
B
)
B∈A
satisfies
naturality.
Given
B
∈
A
and
f
∈
A
(B,
A),
define
x̃
B
(
f
)
=
(X(
f
))(x)
∈
X(B).
(How
else
could
we
possibly
define
it?)
This
makes
sense,
since
X(
f
)
is
a
map
X(A)
→
X(B).
To
prove
naturality,
we
must
show
that
for
any
map
g
B
0
−→
B
in
A
,
the
square
A
(B,
A)
H
A
(g)
=
−◦g
/
A
(B
0
,
A)
x̃
B
0
x̃
B
X(B)
X(g)
/
X(B
0
)
commutes.
To
reduce
clutter,
let
us
write
X(g)
as
Xg,
and
so
on.
Now
for
all
f
∈
A
(B,
A),
we
have
f
_
(X
f
)(x)
/
f
◦
g
_
(X(
f
◦
g))(x)
/
(Xg)((X
f
)(x)),
and
X(
f
◦
g)
=
(Xg)
◦
(X
f
)
by
functoriality,
so
the
square
does
commute.
•
Given
x
∈
X(A),
we
have
to
show
that
x̃
ˆ
=
x,
and
indeed,
x̃
ˆ
=
x̃
A
(1
A
)
=
(X1
A
)(x)
=
1
X(A)
(x)
=
x.
97
4.2
The
Yoneda
lemma
•
Given
α
:
H
A
→
X,
we
have
to
show
that
α̂
˜
=
α.
Two
natural
transformations
are
equal
if
and
only
if
all
their
components
are
equal;
so,
we
have
to
show
that
α̂
˜
=
α
B
for
all
B
∈
A
.
Each
side
of
this
equation
is
a
function
from
B
H
A
(B)
=
A
(B,
A)
to
X(B),
and
two
functions
are
equal
if
and
only
if
they
take
equal
values
at
every
element
of
the
domain;
so,
we
have
to
show
that
α̂
˜
(
f
)
=
α
B
(
f
)
B
for
all
B
∈
A
and
f
:
B
→
A
in
A
.
The
left-hand
side
is
by
definition
α̂
˜
(
f
)
=
(X
f
)(
α̂)
=
(X
f
)(α
A
(1
A
)),
B
so
it
remains
to
prove
that
(X
f
)(α
A
(1
A
))
=
α
B
(
f
).
(4.5)
By
naturality
of
α
(the
only
tool
at
our
disposal),
the
square
A
(A,
A)
H
A
(
f
)
=
−◦
f
α
B
α
A
X(A)
/
A
(B,
A)
Xf
/
X(B)
commutes,
which
when
taken
at
1
A
∈
A
(A,
A)
gives
equation
(4.5).
(The
proof
is
not
over
yet,
but
it
is
worth
pausing
to
consider
the
significance
of
the
fact
that
α̂
˜
=
α.
Since
α̂
is
the
value
of
α
at
1
A
,
this
implies:
A
natural
transformation
H
A
→
X
is
determined
by
its
value
at
1
A
.
Just
how
a
natural
transformation
H
A
→
X
is
determined
by
its
value
at
1
A
is
described
in
equation
(4.5).)
This
establishes
the
bijection
(4.4)
for
each
A
∈
A
and
X
∈
[A
op
,
Set].
We
now
show
that
the
bijection
is
natural
in
A
and
X.
We
employ
two
mildly
labour-saving
devices.
First,
in
principle
we
have
to
prove
naturality
of
both
(
ˆ
)
and
(
˜
),
but
by
Lemma
1.3.11,
it
is
enough
to
prove
naturality
of
just
one
of
them.
We
prove
naturality
of
(
ˆ
).
Second,
by
Exercise
1.3.29,
(
ˆ
)
is
natural
in
the
pair
(A,
X)
if
and
only
if
it
is
natural
in
A
for
each
fixed
X
and
natural
in
X
for
each
fixed
A.
So,
it
remains
to
check
these
two
types
of
naturality.
f
Naturality
in
A
states
that
for
each
X
∈
[A
op
,
Set]
and
B
−→
A
in
A
,
the
98
Representables
square
−◦H
f
[A
op
,
Set](H
A
,
X)
/
[A
op
,
Set](H
B
,
X)
(
ˆ
)
(
ˆ
)
X(A)
/
X(B)
Xf
commutes.
For
α
:
H
A
→
X,
we
have
α
_
/
α
◦
H
f
_
α
◦
H
f
B
(1
B
)
/
(X
f
)(α
A
(1
A
)),
α
A
(1
A
)
so
we
have
to
show
that
α
◦
H
f
B
(1
B
)
=
(X
f
)(α
A
(1
A
)).
Indeed,
α
◦
H
f
B
(1
B
)
=
α
B
((H
f
)
B
(1
B
))
=
α
B
(
f
◦
1
B
)
=
α
B
(
f
)
=
(X
f
)(α
A
(1
A
)),
where
the
first
step
is
by
definition
of
composition
in
[A
op
,
Set],
the
second
is
by
definition
of
H
f
,
and
the
last
is
by
equation
(4.5).
Naturality
in
X
states
that
for
each
A
∈
A
and
map
X
A
op
X
θ
(
6
Set
0
in
[A
,
Set],
the
square
op
[A
op
,
Set](H
A
,
X)
θ◦−
(
ˆ
)
X(A)
commutes.
For
α
:
H
A
→
X,
we
have
α
_
α
A
(1
A
)
/
[A
op
,
Set](H
A
,
X
0
)
(
ˆ
)
θ
A
/
X
0
(A)
/
θ
◦
α
_
(θ
◦
α)
A
(1
A
)
/
θ
A
(α
A
(1
A
)),
and
(θ
◦
α)
A
=
θ
A
◦
α
A
by
definition
of
composition
in
[A
op
,
Set],
so
the
square
does
commute.
This
completes
the
proof.
99
4.3
Consequences
of
the
Yoneda
lemma
Exercises
4.2.2
State
the
dual
of
the
Yoneda
lemma.
4.2.3
One
way
to
understand
the
Yoneda
lemma
is
to
examine
some
special
cases.
Here
we
consider
one-object
categories.
Let
M
be
a
monoid.
The
underlying
set
of
M
can
be
given
a
right
M-action
by
multiplication:
x
·
m
=
xm
for
all
x,
m
∈
M.
This
M-set
is
called
the
right
regular
representation
of
M.
Let
us
write
it
as
M.
(a)
When
M
is
regarded
as
a
one-object
category,
functors
M
op
→
Set
corre-
spond
to
right
M-sets
(Example
1.2.14).
Show
that
the
M-set
correspond-
ing
to
the
unique
representable
functor
M
op
→
Set
is
the
right
regular
representation.
(b)
Now
let
X
be
any
right
M-set.
Show
that
for
each
x
∈
X,
there
is
a
unique
map
α
:
M
→
X
of
right
M-sets
such
that
α(1)
=
x.
Deduce
that
there
is
a
bijection
between
{maps
M
→
X
of
right
M-sets}
and
X.
(c)
Deduce
the
Yoneda
lemma
for
one-object
categories.
4.3
Consequences
of
the
Yoneda
lemma
The
Yoneda
lemma
is
fundamental
in
category
theory.
Here
we
look
at
three
important
consequences.
Notation
4.3.1
isomorphism.
∼
An
arrow
decorated
with
a
∼,
as
in
A
−→
B,
denotes
an
A
representation
is
a
universal
element
Corollary
4.3.2
Let
A
be
a
locally
small
category
and
X
:
A
op
→
Set.
Then
a
representation
of
X
consists
of
an
object
A
∈
A
together
with
an
element
u
∈
X(A)
such
that:
for
each
B
∈
A
and
x
∈
X(B),
there
is
a
unique
map
x̄
:
B
→
A
such
that
(X
x̄)(u)
=
x.
(4.6)
To
clarify
the
statement,
first
recall
that
by
definition,
a
representation
of
X
is
∼
an
object
A
∈
A
together
with
a
natural
isomorphism
α
:
H
A
−→
X.
Corol-
lary
4.3.2
states
that
such
pairs
(A,
α)
are
in
natural
bijection
with
pairs
(A,
u)
satisfying
condition
(4.6).
Pairs
(B,
x)
with
B
∈
A
and
x
∈
X(B)
are
sometimes
called
elements
of
the
presheaf
X.
(Indeed,
the
Yoneda
lemma
tells
us
that
x
amounts
to
a
gen-
eralized
element
of
X
of
shape
H
B
.)
An
element
u
satisfying
condition
(4.6)
100
Representables
is
sometimes
called
a
universal
element
of
X.
So,
Corollary
4.3.2
says
that
a
representation
of
a
presheaf
X
amounts
to
a
universal
element
of
X.
Proof
By
the
Yoneda
lemma,
we
have
only
to
show
that
for
A
∈
A
and
u
∈
X(A),
the
natural
transformation
ũ
:
H
A
→
X
is
an
isomorphism
if
and
only
if
(4.6)
holds.
(Here
we
are
using
the
notation
introduced
in
the
proof
of
the
Yoneda
lemma.)
Now,
ũ
is
an
isomorphism
if
and
only
if
for
all
B
∈
A
,
the
function
ũ
B
:
H
A
(B)
=
A
(B,
A)
→
X(B)
is
a
bijection,
if
and
only
if
for
all
B
∈
A
and
x
∈
X(B),
there
is
a
unique
x̄
∈
A
(B,
A)
such
that
ũ
B
(
x̄)
=
x.
But
ũ
B
(
x̄)
=
(X
x̄)(u),
so
this
is
exactly
condition
(4.6).
Our
examples
will
use
the
dual
form,
for
covariant
set-valued
functors:
Corollary
4.3.3
Let
A
be
a
locally
small
category
and
X
:
A
→
Set.
Then
a
representation
of
X
consists
of
an
object
A
∈
A
together
with
an
element
u
∈
X(A)
such
that:
for
each
B
∈
A
and
x
∈
X(B),
there
is
a
unique
map
x̄
:
A
→
B
such
that
(X
x̄)(u)
=
x.
Proof
(4.7)
Follows
immediately
by
duality.
Example
4.3.4
Fix
a
set
S
and
consider
the
functor
X
=
Set(S
,
U(−))
:
Vect
k
V
→
Set
7
→
Set(S
,
U(V)).
Here
are
two
familiar
(and
true!)
statements
about
X:
(a)
there
exist
a
vector
space
F(S
)
and
an
isomorphism
Vect
k
(F(S
),
V)
Set(S
,
U(V))
(4.8)
natural
in
V
∈
Vect
k
(Example
2.1.3(a));
(b)
there
exist
a
vector
space
F(S
)
and
a
function
u
:
S
→
U(F(S
))
such
that:
for
each
vector
space
V
and
function
f
:
S
→
U(V),
there
is
a
unique
linear
map
f
¯
:
F(S
)
→
V
such
that
S
/
U(F(S
))
u
f
commutes
U(
f
¯
)
#
U(V)
101
4.3
Consequences
of
the
Yoneda
lemma
(as
in
the
introduction
to
Section
2.3,
where
u
was
called
by
its
usual
name,
η
S
).
Each
of
these
two
statements
says
that
X
is
representable.
Statement
(a)
says
that
there
is
an
isomorphism
X(V)
Vect(F(S
),
V)
natural
in
V,
that
is,
an
iso-
morphism
X
H
F(S
)
.
So
X
is
representable,
by
definition
of
representability.
Statement
(b)
says
that
u
∈
X(F(S
))
satisfies
condition
(4.7).
So
X
is
repre-
sentable,
by
Corollary
4.3.3.
You
will
have
noticed
that
the
first
way
of
saying
that
X
is
representable
is
substantially
shorter
than
the
second.
Indeed,
it
is
clear
that
if
the
situation
of
(b)
holds
then
there
is
an
isomorphism
∼
Vect
k
(F(S
),
V)
−→
Set(S
,
U(V))
natural
in
V,
defined
by
g
7→
U(g)
◦
u.
But
it
looks
at
first
as
if
(b)
says
rather
more
than
(a),
since
it
states
that
the
two
functors
are
not
only
naturally
isomorphic,
but
naturally
isomorphic
in
a
rather
special
way.
Corollary
4.3.3
tells
us
that
this
is
an
illusion:
all
natural
isomorphisms
(4.8)
arise
in
this
way.
It
is
the
word
‘natural’
in
(a)
that
hides
the
explicit
detail.
Example
4.3.5
The
same
can
be
said
for
any
other
adjunction
A
o
F
⊥
/
B
.
G
Fix
A
∈
A
and
put
X
=
A
(A,
G(−))
:
B
→
Set.
Then
X
is
representable,
and
this
can
be
expressed
in
either
of
the
following
ways:
(a)
A
(A,
G(B))
B(F(A),
B)
naturally
in
B;
in
other
words,
X
H
F(A)
(as
in
Lemma
4.1.10);
(b)
the
unit
map
η
A
:
A
→
G(F(A))
is
an
initial
object
of
the
comma
category
(A
⇒
G);
that
is,
η
A
∈
X(F(A))
satisfies
condition
(4.7).
This
observation
can
be
developed
into
an
alternative
proof
of
Theorem
2.3.6,
the
reformulation
of
adjointness
in
terms
of
initial
objects.
Example
4.3.6
For
any
group
G
and
element
x
∈
G,
there
is
a
unique
ho-
momorphism
φ
:
Z
→
G
such
that
φ(1)
=
x.
This
means
that
1
∈
U(Z)
is
a
universal
element
of
the
forgetful
functor
U
:
Grp
→
Set;
in
other
words,
con-
dition
(4.7)
holds
when
A
=
Grp,
X
=
U,
A
=
Z
and
u
=
1.
So
1
∈
U(Z)
∼
gives
a
representation
H
Z
−→
U
of
U.
On
the
other
hand,
the
same
is
true
with
−1
in
place
of
1.
The
isomorphisms
102
Representables
∼
H
Z
−→
U
coming
from
1
and
−1
are
not
equal,
because
Corollary
4.3.3
pro-
vides
a
one-to-one
correspondence
between
universal
elements
and
represen-
tations.
The
Yoneda
embedding
Here
is
a
second
corollary
of
the
Yoneda
lemma.
Corollary
4.3.7
For
any
locally
small
category
A
,
the
Yoneda
embedding
H
•
:
A
→
[A
op
,
Set]
is
full
and
faithful.
Informally,
this
says
that
for
A,
A
0
∈
A
,
a
map
H
A
→
H
A
0
of
presheaves
is
the
same
thing
as
a
map
A
→
A
0
in
A
.
Proof
We
have
to
show
that
for
each
A,
A
0
∈
A
,
the
function
A
(A,
A
0
)
→
f
7→
[A
op
,
Set](H
A
,
H
A
0
)
H
f
(4.9)
is
bijective.
By
the
Yoneda
lemma
(taking
‘X’
to
be
H
A
0
),
the
function
(
˜
)
:
H
A
0
(A)
→
[A
op
,
Set](H
A
,
H
A
0
)
(4.10)
is
bijective,
so
it
is
enough
to
prove
that
the
functions
(4.9)
and
(4.10)
are
equal.
Thus,
given
f
:
A
→
A
0
,
we
have
to
prove
that
f
˜
=
H
f
,
or
equivalently,
c
f
=
f
.
And
indeed,
H
c
f
=
(H
f
)
A
(1
A
)
=
f
◦
1
A
=
f,
H
as
required.
In
mathematics
at
large,
the
word
‘embedding’
is
used
(sometimes
infor-
mally)
to
mean
a
map
A
→
B
that
makes
A
isomorphic
to
its
image
in
B.
For
example,
an
injection
of
sets
i
:
A
→
B
might
be
called
an
embedding,
because
it
provides
a
bijection
between
A
and
the
subset
iA
of
B.
Similarly,
a
map
i
:
A
→
B
of
topological
spaces
might
be
called
an
embedding
if
it
is
a
homeomorphism
to
its
image,
so
that
A
iA.
Corollary
1.3.19
tells
us
that
in
category
theory,
a
full
and
faithful
functor
A
→
B
can
reasonably
be
called
an
embedding,
as
it
makes
A
equivalent
to
a
full
subcategory
of
B.
In
the
case
at
hand,
the
Yoneda
embedding
H
•
:
A
→
[A
op
,
Set]
embeds
A
into
its
own
presheaf
category
(Figure
4.1).
So,
A
is
equivalent
to
the
full
subcategory
of
[A
op
,
Set]
whose
objects
are
the
representables.
4.3
Consequences
of
the
Yoneda
lemma
A
103
[A
op
,
Set]
Figure
4.1
A
category
A
embedded
into
its
presheaf
category.
In
general,
full
subcategories
are
the
easiest
subcategories
to
handle.
For
instance,
given
objects
A
and
A
0
of
a
full
subcategory,
we
can
speak
unam-
biguously
of
the
‘maps’
from
A
to
A
0
;
it
makes
no
difference
whether
this
is
understood
to
mean
maps
in
the
subcategory
or
maps
in
the
whole
category.
Similarly,
we
can
speak
unambiguously
of
isomorphism
of
objects
of
the
sub-
category,
as
in
the
following
lemma.
Lemma
4.3.8
Then:
Let
J
:
A
→
B
be
a
full
and
faithful
functor
and
A,
A
0
∈
A
.
(a)
a
map
f
in
A
is
an
isomorphism
if
and
only
if
the
map
J(
f
)
in
B
is
an
isomorphism;
(b)
for
any
isomorphism
g
:
J(A)
→
J(A
0
)
in
B,
there
is
a
unique
isomorphism
f
:
A
→
A
0
in
A
such
that
J(
f
)
=
g;
(c)
the
objects
A
and
A
0
of
A
are
isomorphic
if
and
only
if
the
objects
J(A)
and
J(A
0
)
of
B
are
isomorphic.
Proof
Exercise
4.3.15.
Example
4.3.9
In
Example
4.3.6,
we
considered
the
representations
of
the
forgetful
functor
U
:
Grp
→
Set,
and
found
two
different
isomorphisms
∼
H
Z
−→
U.
Did
we
find
all
of
them?
∼
Since
H
Z
U,
there
are
as
many
isomorphisms
H
Z
−→
U
as
there
are
∼
isomorphisms
H
Z
−→
H
Z
.
By
Corollary
4.3.7
and
Lemma
4.3.8(b),
there
are
∼
as
many
of
these
as
there
are
group
isomorphisms
Z
−→
Z.
There
are
precisely
two
such
(corresponding
to
the
two
generators
±1
of
Z),
so
we
did
indeed
find
∼
all
the
isomorphisms
H
Z
−→
U.
Differently
put,
there
are
exactly
two
universal
elements
of
U(Z).
In
Section
6.2,
we
will
see
that
every
presheaf
can
be
built
from
representa-
bles,
in
very
roughly
the
same
way
that
every
positive
integer
can
be
built
from
primes.
104
Representables
A
A
A
0
B
1
B
2
B
3
Figure
4.2
If
A
(B,
A)
A
(B,
A
0
)
naturally
in
B,
then
A
A
0
.
Isomorphism
of
representables
In
Exercise
4.1.27,
you
were
asked
to
prove
directly
that
if
H
A
H
A
0
then
A
A
0
.
The
proof
contains
all
the
main
ideas
in
the
proof
of
the
Yoneda
lemma.
The
result
itself
can
also
be
deduced
from
the
Yoneda
lemma,
as
follows.
Corollary
4.3.10
Let
A
be
a
locally
small
category
and
A,
A
0
∈
A
.
Then
0
H
A
H
A
0
⇐⇒
A
A
0
⇐⇒
H
A
H
A
.
Proof
By
duality,
it
is
enough
to
prove
the
first
‘⇐⇒’.
This
follows
from
Corollary
4.3.7
and
Lemma
4.3.8(c).
Since
functors
always
preserve
isomorphism
(Exercise
1.2.21),
the
force
of
this
statement
is
that
H
A
H
A
0
=⇒
A
A
0
.
In
other
words,
if
A
(B,
A)
A
(B,
A
0
)
naturally
in
B,
then
A
A
0
.
Thinking
of
A
(B,
A)
as
‘A
viewed
from
B’,
the
corollary
tells
us
that
two
objects
are
the
same
if
and
only
if
they
look
the
same
from
all
viewpoints
(Figure
4.2).
(If
it
looks
like
a
duck,
walks
like
a
duck,
and
quacks
like
a
duck,
then
it
probably
is
a
duck.)
Example
4.3.11
Consider
Corollary
4.3.10
in
the
case
A
=
Grp.
Take
two
groups
A
and
A
0
,
and
suppose
someone
tells
us
that
A
and
A
0
‘look
the
same
from
B’
(meaning
that
H
A
(B)
H
A
0
(B))
for
all
groups
B.
Then,
for
instance:
•
H
A
(1)
H
A
0
(1),
where
1
is
the
trivial
group.
But
H
A
(1)
=
Grp(1,
A)
is
a
one-element
set,
as
is
H
A
0
(1),
no
matter
what
A
and
A
0
are.
So
this
tells
us
nothing
at
all.
•
H
A
(Z)
H
A
0
(Z).
We
know
that
H
A
(Z)
is
the
underlying
set
of
A,
and
sim-
ilarly
for
A
0
.
So
A
and
A
0
have
isomorphic
underlying
sets.
But
for
all
we
know
so
far,
they
might
have
entirely
different
group
structures.
4.3
Consequences
of
the
Yoneda
lemma
105
•
H
A
(Z/pZ)
H
A
0
(Z/pZ)
for
every
prime
p,
so
by
Example
4.1.5,
A
and
A
0
have
the
same
number
of
elements
of
each
prime
order.
Each
of
these
isomorphisms
gives
only
partial
information
about
the
similar-
ity
of
A
and
A
0
.
But
if
we
know
that
H
A
(B)
H
A
0
(B)
for
all
groups
B,
and
naturally
in
B,
then
A
A
0
.
Example
4.3.12
set
A,
we
have
The
category
of
sets
is
very
unusual
in
this
respect.
For
any
A
Set(1,
A)
=
H
A
(1),
so
H
A
(1)
H
A
0
(1)
implies
A
A
0
.
In
other
words,
two
objects
of
Set
are
the
same
if
they
look
the
same
from
the
point
of
view
of
the
one-element
set.
This
is
a
familiar
feature
of
sets:
the
only
thing
that
matters
about
a
set
is
its
elements!
For
a
general
category,
Corollary
4.3.10
tells
us
that
two
objects
are
the
same
if
they
have
the
same
generalized
elements
of
all
shapes.
But
the
category
of
sets
has
a
special
property:
if
I
choose
an
object
and
tell
you
only
what
its
generalized
elements
of
shape
1
are,
then
you
can
deduce
exactly
what
my
object
must
be.
Example
4.3.13
Let
G
:
B
→
A
be
a
functor,
and
suppose
that
both
F
and
F
0
are
left
adjoint
to
G.
Then
for
each
A
∈
A
,
we
have
B(F(A),
B)
A
(A,
G(B))
B(F
0
(A),
B)
0
naturally
in
B
∈
B,
so
H
F(A)
H
F
(A)
,
so
F(A)
F
0
(A)
by
Corollary
4.3.10.
In
fact,
this
isomorphism
is
natural
in
A,
so
that
F
F
0
.
This
shows
that
left
adjoints
are
unique,
as
claimed
in
Remark
2.1.2(d).
Dually,
right
adjoints
are
unique.
See
also
Exercise
4.3.18.
Example
4.3.14
Corollary
4.3.10
implies
that
if
a
set-valued
functor
is
iso-
0
morphic
to
both
H
A
and
H
A
then
A
A
0
.
So
the
functor
determines
the
repre-
senting
object,
if
one
exists.
For
instance,
take
the
functor
Bilin(U,
V;
−)
:
Vect
k
→
Set
of
Example
4.1.9.
Corollary
4.3.10
implies
that
up
to
isomorphism,
there
is
at
most
one
vector
space
T
such
that
Bilin(U,
V;
W)
Vect
k
(T,
W)
naturally
in
W.
It
can
be
shown
that
there
does,
in
fact,
exist
such
a
vector
space
T
.
Since
all
such
spaces
T
are
isomorphic,
it
is
legitimate
to
refer
to
any
of
them
as
the
tensor
product
of
U
and
V.
106
Representables
Exercises
4.3.15
Prove
Lemma
4.3.8.
4.3.16
Let
A
be
a
locally
small
category.
Prove
each
of
the
following
state-
ments
directly
(without
using
the
Yoneda
lemma).
(a)
H
•
:
A
→
[A
op
,
Set]
is
faithful.
(b)
H
•
is
full.
(c)
Given
A
∈
A
and
a
presheaf
X
on
A
,
if
X(A)
has
an
element
u
that
is
universal
in
the
sense
of
Corollary
4.3.2,
then
X
H
A
.
4.3.17
Interpret
the
theory
of
Chapter
4
in
the
case
where
the
category
A
is
discrete.
For
example,
what
do
presheaves
look
like,
and
which
ones
are
representable?
What
does
the
Yoneda
lemma
tell
us?
Does
its
proof
become
any
shorter?
What
about
the
corollaries
of
the
Yoneda
lemma?
4.3.18
Let
B
be
a
category
and
J
:
C
→
D
a
functor.
There
is
an
induced
functor
J
◦
−
:
[B,
C
]
→
[B,
D]
defined
by
composition
with
J.
(a)
Show
that
if
J
is
full
and
faithful
then
so
is
J
◦
−.
(b)
Deduce
that
if
J
is
full
and
faithful
and
G,
G
0
:
B
→
C
with
J
◦
G
J
◦
G
0
then
G
G
0
.
(c)
Now
deduce
that
right
adjoints
are
unique:
if
F
:
A
→
B
and
G,
G
0
:
B
→
A
with
F
a
G
and
F
a
G
0
then
G
G
0
.
(Hint:
the
Yoneda
embedding
is
full
and
faithful.)
5
Limits
Limits,
and
the
dual
concept,
colimits,
provide
our
third
approach
to
the
idea
of
universal
property.
Adjointness
is
about
the
relationships
between
categories.
Representability
is
a
property
of
set-valued
functors.
Limits
are
about
what
goes
on
inside
a
category.
The
concept
of
limit
unifies
many
familiar
constructions
in
mathematics.
Whenever
you
meet
a
method
for
taking
some
objects
and
maps
in
a
category
and
constructing
a
new
object
out
of
them,
there
is
a
good
chance
that
you
are
looking
at
either
a
limit
or
a
colimit.
For
instance,
in
group
theory,
we
can
take
a
homomorphism
between
two
groups
and
form
its
kernel,
which
is
a
new
group.
This
construction
is
an
example
of
a
limit
in
the
category
of
groups.
Or,
we
might
take
two
natural
numbers
and
form
their
lowest
common
multiple.
This
is
an
example
of
a
colimit
in
the
poset
of
natural
numbers,
ordered
by
divisibility.
5.1
Limits:
definition
and
examples
The
definition
of
limit
is
very
general.
We
build
up
to
it
by
first
examining
some
particularly
useful
types
of
limit:
products,
equalizers,
and
pullbacks.
Products
Let
X
and
Y
be
sets.
The
familiar
cartesian
product
X×Y
is
characterized
by
the
property
that
an
element
of
X
×
Y
is
an
element
of
X
together
with
an
element
of
Y.
Since
elements
are
just
maps
from
1,
this
says
that
a
map
1
→
X
×
Y
amounts
to
a
map
1
→
X
together
with
a
map
1
→
Y.
A
little
thought
reveals
that
the
same
is
true
when
1
is
replaced
throughout
107
108
Limits
by
any
set
A
whatsoever.
(In
other
words,
a
generalized
element
of
X
×
Y
of
shape
A
amounts
to
a
generalized
element
of
X
of
shape
A
together
with
a
generalized
element
of
Y
of
shape
A.)
The
bijection
between
maps
A
→
X
×
Y
and
pairs
of
maps
(A
→
X,
A
→
Y)
is
given
by
composing
with
the
projection
maps
X
x
p
1
←−
→
7
X
×
Y
(x,
y)
p
2
−→
7→
Y
y.
This
suggests
the
following
definition.
Definition
5.1.1
Let
A
be
a
category
and
X,
Y
∈
A
.
A
product
of
X
and
Y
consists
of
an
object
P
and
maps
P
p
1
p
2
X
Y
with
the
property
that
for
all
objects
and
maps
A
f
1
f
2
X
(5.1)
Y
in
A
,
there
exists
a
unique
map
f
¯
:
A
→
P
such
that
A
f
¯
P
f
1
X
p
1
f
2
p
2
(5.2)
Y
commutes.
The
maps
p
1
and
p
2
are
called
the
projections.
Remarks
5.1.2
(a)
Products
do
not
always
exist.
For
example,
if
A
is
the
discrete
two-object
category
X•
•Y
5.1
Limits:
definition
and
examples
109
then
X
and
Y
do
not
have
a
product.
But
when
objects
X
and
Y
of
a
category
do
have
a
product,
it
is
unique
up
to
isomorphism.
(This
can
be
proved
directly,
much
as
in
Lemma
2.1.8.
It
also
follows
from
Corollary
6.1.2.)
This
justifies
talking
about
the
product
of
X
and
Y.
(b)
Strictly
speaking,
the
product
consists
of
the
object
P
together
with
the
projections
p
1
and
p
2
.
But
informally,
we
often
refer
to
P
alone
as
the
product
of
X
and
Y.
We
write
P
as
X
×
Y.
Example
5.1.3
Any
two
sets
X
and
Y
have
a
product
in
Set.
It
is
the
usual
cartesian
product
X
×
Y,
equipped
with
the
usual
projection
maps
p
1
and
p
2
.
Let
us
check
that
this
really
is
a
product
in
the
sense
of
Definition
5.1.1.
Take
sets
and
functions
as
in
diagram
(5.1).
Define
f
¯
:
A
→
X
×
Y
by
f
¯
(a)
=
(
f
1
(a),
f
2
(a)).
Then
p
i
◦
f
¯
=
f
i
for
i
=
1,
2;
that
is,
diagram
(5.2)
commutes
with
P
=
X×Y.
Moreover,
this
is
the
only
map
making
diagram
(5.2)
commute.
For
suppose
that
f
ˆ
:
A
→
X
×
Y,
in
place
of
f
¯
,
also
makes
(5.2)
commute.
Let
a
∈
A,
and
write
f
ˆ
(a)
as
(x,
y).
Then
f
1
(a)
=
p
1
(
f
ˆ
(a))
=
p
1
(x,
y)
=
x,
and
similarly,
f
2
(a)
=
y.
Hence
f
ˆ
(a)
=
(
f
1
(a),
f
2
(a))
=
f
¯
(a)
for
all
a
∈
A,
giving
f
ˆ
=
f
¯
,
as
required.
In
general,
in
any
category,
the
map
f
¯
of
diagram
(5.2)
is
usually
written
as
(
f
1
,
f
2
).
Example
5.1.4
In
the
category
of
topological
spaces,
any
two
objects
X
and
Y
have
a
product.
It
is
the
set
X
×
Y
equipped
with
the
product
topology
and
the
standard
projection
maps.
The
product
topology
is
deliberately
designed
so
that
a
function
A
t
→
7
→
X
×
Y
(x(t),
y(t))
is
continuous
if
and
only
if
it
is
continuous
in
each
coordinate
(that
is
to
say,
both
functions
t
7→
x(t),
t
7→
y(t)
are
continuous).
This
holds
for
any
space
A,
but
the
idea
is
perhaps
at
its
most
intuitively
appealing
when
A
=
R
and
we
think
of
t
as
a
time
parameter.
A
closely
related
statement
is
that
the
product
topology
is
the
smallest
topol-
ogy
on
X
×
Y
for
which
the
projections
are
continuous.
Here
‘smallest’
means
that
for
any
other
topology
T
on
X
×
Y
such
that
p
1
and
p
2
are
continuous,
every
subset
of
X
×
Y
open
in
the
product
topology
is
also
open
in
T
.
Thus,
110
Limits
to
define
the
product
topology,
we
declare
just
enough
sets
to
be
open
that
the
projections
are
continuous.
Example
5.1.5
Now
let
X
and
Y
be
vector
spaces.
We
can
form
their
direct
sum,
X
⊕
Y,
whose
elements
can
be
written
as
either
(x,
y)
or
x
+
y
(with
x
∈
X
and
y
∈
Y),
according
to
taste.
There
are
linear
projection
maps
X
⊕
Y
p
1
X
|
;
(x,
y)
p
2
"
Y
!
}
x
y.
It
can
be
shown
that
X
⊕
Y,
together
with
p
1
and
p
2
,
is
the
product
of
X
and
Y
in
the
category
of
vector
spaces
(Exercise
5.1.33).
Examples
5.1.6
(Elements
of
ordered
sets)
mum
min{x,
y}
satisfies
min{x,
y}
≤
x,
(a)
Let
x,
y
∈
R.
Their
mini-
min{x,
y}
≤
y
and
has
the
further
property
that
whenever
a
∈
R
with
a
≤
x,
a
≤
y,
we
have
a
≤
min{x,
y}.
This
means
exactly
that
when
the
poset
(R,
≤)
is
viewed
as
a
category,
the
product
of
x,
y
∈
R
is
min{x,
y}.
The
definition
of
product
simplifies
when
interpreted
in
a
poset,
since
all
diagrams
com-
mute.
(b)
Fix
a
set
S
.
Let
X,
Y
∈
P(S
).
Then
X
∩
Y
satisfies
X
∩
Y
⊆
X,
X
∩
Y
⊆
Y
and
has
the
further
property
that
whenever
A
∈
P(S
)
with
A
⊆
X,
A
⊆
Y,
we
have
A
⊆
X
∩
Y.
This
means
that
X
∩
Y
is
the
product
of
X
and
Y
in
the
poset
(P(S
),
⊆)
regarded
as
a
category.
(c)
Let
x,
y
∈
N.
Their
greatest
common
divisor
gcd(x,
y)
satisfies
gcd(x,
y)
|
x,
gcd(x,
y)
|
y
(it’s
a
common
divisor!)
and
has
the
further
property
that
whenever
a
∈
N
with
a
|
x,
a
|
y,
5.1
Limits:
definition
and
examples
111
we
have
a
|
gcd(x,
y).
This
means
that
gcd(x,
y)
is
the
product
of
x
and
y
in
the
poset
(N,
|)
regarded
as
a
category.
Generally,
let
(A,
≤)
be
a
poset
and
x,
y
∈
A.
A
lower
bound
for
x
and
y
is
an
element
a
∈
A
such
that
a
≤
x
and
a
≤
y.
A
greatest
lower
bound
or
meet
of
x
and
y
is
a
lower
bound
z
for
x
and
y
with
the
further
property
that
whenever
a
is
a
lower
bound
for
x
and
y,
we
have
a
≤
z.
When
a
poset
is
regarded
as
a
category,
meets
are
exactly
products.
They
do
not
always
exist,
but
when
they
do,
they
are
unique.
The
meet
of
x
and
y
is
usually
written
as
x
∧
y
rather
than
x
×
y.
Thus,
in
the
three
examples
above,
x
∧
y
=
min{x,
y},
X
∧
Y
=
X
∩
Y,
x
∧
y
=
gcd(x,
y),
the
second
example
being
the
origin
of
the
notation.
We
have
been
discussing
products
X
×
Y
of
two
objects,
so-called
binary
products.
But
there
is
no
reason
to
stick
to
two.
We
can
just
as
well
talk
about
products
X
×Y
×Z
of
three
objects,
or
of
infinitely
many
objects.
The
definition
changes
in
the
most
obvious
way:
Definition
5.1.7
Let
A
be
a
category,
I
a
set,
and
(X
i
)
i∈I
a
family
of
objects
of
A
.
A
product
of
(X
i
)
i∈I
consists
of
an
object
P
and
a
family
of
maps
p
i
P
−→
X
i
i∈I
with
the
property
that
for
all
objects
A
and
families
of
maps
f
i
A
−→
X
i
i∈I
(5.3)
there
exists
a
unique
map
f
¯
:
A
→
P
such
that
p
i
◦
f
¯
=
f
i
for
all
i
∈
I.
Remarks
5.1.2
apply
equally
to
this
definition.
When
the
product
P
exists,
Q
we
write
P
as
i∈I
X
i
and
the
map
f
¯
as
(
f
i
)
i∈I
.
We
call
the
maps
f
i
the
com-
ponents
of
the
map
(
f
i
)
i∈I
.
Taking
I
to
be
a
two-element
set,
we
recover
the
special
case
of
binary
products.
Example
5.1.8
In
ordered
sets,
the
extension
from
binary
to
arbitrary
prod-
ucts
works
in
the
obvious
way:
given
an
ordered
set
(A,
≤),
a
lower
bound
for
a
family
(x
i
)
i∈I
of
elements
is
an
element
a
∈
A
such
that
a
≤
x
i
for
all
i,
and
a
greatest
lower
bound
or
meet
of
the
family
is
a
lower
bound
greater
than
any
V
other,
written
as
i∈I
x
i
.
These
are
the
products
in
(A,
≤).
For
example,
in
R
with
its
usual
ordering,
the
meet
of
a
family
(x
i
)
i∈I
is
inf{x
i
|
i
∈
I}
(and
one
exists
if
and
only
if
the
other
does).
Example
5.1.9
What
happens
to
the
definition
of
product
when
the
indexing
set
I
is
empty?
Let
A
be
a
category.
In
general,
an
I-indexed
family
(X
i
)
i∈I
of
112
Limits
objects
of
A
is
a
function
I
→
ob(A
).
When
I
is
empty,
there
is
exactly
one
such
function.
In
other
words,
there
is
exactly
one
family
(X
i
)
i∈∅
,
the
empty
family.
Similarly,
when
I
is
empty,
there
is
exactly
one
family
(5.3)
for
any
given
object
A.
A
product
of
the
empty
family
therefore
consists
of
an
object
P
of
A
such
that
for
each
object
A
of
A
,
there
exists
a
unique
map
f
¯
:
A
→
P.
(The
condi-
tion
‘p
i
◦
f
¯
=
f
i
for
all
i
∈
I’
holds
trivially.)
In
other
words,
a
product
of
the
empty
family
is
exactly
a
terminal
object.
We
have
been
writing
1
for
terminal
objects,
which
was
justified
by
the
fact
that
in
categories
such
as
Set,
Top,
Ring
and
Grp,
the
terminal
object
has
one
element.
But
we
have
just
seen
that
the
terminal
object
is
the
product
of
no
things,
which
in
the
context
of
elementary
arithmetic
is
the
number
1.
This
is
a
second,
related,
reason
for
the
notation.
Example
5.1.10
Take
an
object
X
of
a
category
A
,
and
a
set
I.
There
is
Q
a
constant
family
(X)
i∈I
.
Its
product
i∈I
X,
if
it
exists,
is
written
as
X
I
and
called
a
power
of
X.
We
met
powers
in
Set
in
Section
3.1.
When
X
is
a
set,
X
I
is
the
set
of
functions
from
I
to
X,
also
written
as
Set(I,
X).
Equalizers
To
define
our
second
type
of
limit,
we
need
a
preliminary
piece
of
terminology:
a
fork
in
a
category
consists
of
objects
and
maps
A
f
s
/
X
t
/
/
Y
(5.4)
such
that
s
f
=
t
f
.
Definition
5.1.11
Let
A
be
a
category
and
let
X
s
t
/
/
Y
be
objects
and
maps
i
in
A
.
An
equalizer
of
s
and
t
is
an
object
E
together
with
a
map
E
−→
X
such
that
s
i
/
/
Y
/
X
E
t
is
a
fork,
and
with
the
property
that
for
any
fork
(5.4),
there
exists
a
unique
map
f
¯
:
A
→
E
such
that
A
f
f
¯
E
(5.5)
i
/
X
5.1
Limits:
definition
and
examples
113
commutes.
Remarks
5.1.2
on
products
apply
to
equalizers
too.
Example
5.1.12
We
have
already
met
equalizers
in
Set
(Section
3.1).
They
really
are
equalizers
in
the
sense
of
Definition
5.1.11.
Indeed,
take
sets
and
s
/
/
Y,
write
functions
X
t
E
=
{x
∈
X
|
s(x)
=
t(x)},
and
write
i
:
E
→
X
for
the
inclusion.
Then
si
=
ti,
so
we
have
a
fork,
and
one
can
check
that
it
is
universal
among
all
forks
on
s
and
t.
An
equalizer
describes
the
set
of
solutions
of
a
single
equation,
but
by
com-
bining
equalizers
with
products,
we
can
also
describe
the
solution-set
of
any
system
of
simultaneous
equations.
Take
a
set
Λ
and
a
family
s
λ
X
t
λ
/
/
Y
λ
λ∈Λ
of
pairs
of
maps
in
Set.
Then
the
solution-set
{x
∈
X
|
s
λ
(x)
=
t
λ
(x)
for
all
λ
∈
Λ}
is
the
equalizer
of
the
functions
(s
λ
)
λ∈Λ
X
(t
λ
)
λ∈Λ
/
/
Y
Y
λ
λ∈Λ
(using
the
notation
introduced
after
Definition
5.1.7).
To
see
this,
observe
that
for
x
∈
X,
(s
λ
)
λ∈Λ
(x)
=
(t
λ
)
λ∈Λ
(x)
⇐⇒
s
λ
(x)
λ∈Λ
=
t
λ
(x)
λ∈Λ
⇐⇒
s
λ
(x)
=
t
λ
(x)
for
all
λ
∈
Λ,
as
required.
Example
5.1.13
Take
continuous
maps
X
s
t
/
/
Y
between
topological
spaces.
We
can
form
their
equalizer
E
in
the
category
of
sets,
with
inclusion
map
i
:
E
→
X,
say.
Since
E
is
a
subset
of
the
space
X,
it
acquires
the
subspace
topology
from
X,
and
i
is
then
continuous.
This
space
E,
together
with
i,
is
the
equalizer
of
s
and
t.
114
Limits
Showing
this
amounts
to
showing
that
for
any
fork
(5.4)
in
Top,
the
in-
duced
function
f
¯
is
continuous.
This
follows
from
the
definition
of
the
sub-
space
topology,
which
is
the
smallest
topology
such
that
the
inclusion
map
is
continuous.
Compare
the
remarks
on
products
in
Example
5.1.4.
Example
5.1.14
Let
θ
:
G
→
H
be
a
homomorphism
of
groups.
As
in
Exam-
ple
0.8,
the
homomorphism
θ
gives
rise
to
a
fork
ker
θ
ι
θ
/
G
ε
/
/
H
where
ι
is
the
inclusion
and
ε
is
the
trivial
homomorphism.
This
is
an
equalizer
in
Grp.
Showing
this
amounts
to
showing
that
the
map
that
we
have
been
calling
f
¯
is
a
homomorphism,
which
is
left
to
the
reader.
Thus,
kernels
are
a
special
case
of
equalizers.
Example
5.1.15
Let
V
s
t
/
/
W
be
linear
maps
between
vector
spaces.
There
is
a
linear
map
t
−
s
:
V
→
W,
and
the
equalizer
of
s
and
t
in
the
cat-
egory
of
vector
spaces
is
the
space
ker(t
−
s)
together
with
the
inclusion
map
ker(t
−
s)
,→
V.
Pullbacks
We
explore
one
more
type
of
limit
before
formulating
the
general
definition.
Definition
5.1.16
Let
A
be
a
category,
and
take
objects
and
maps
Y
X
/
Z
s
(5.6)
t
in
A
.
A
pullback
of
this
diagram
is
an
object
P
∈
A
together
with
maps
p
1
:
P
→
X
and
p
2
:
P
→
Y
such
that
P
p
1
X
p
2
s
/
Y
/
Z
t
(5.7)
5.1
Limits:
definition
and
examples
115
commutes,
and
with
the
property
that
for
any
commutative
square
f
2
A
f
1
X
s
/
Y
/
Z
(5.8)
t
in
A
,
there
is
a
unique
map
f
¯
:
A
→
P
such
that
A
f
2
f
¯
P
f
1
p
2
p
1
X
s
"
/
Y
/
Z
(5.9)
t
commutes.
(For
(5.9)
to
commute
means
only
that
p
1
f
¯
=
f
1
and
p
2
f
¯
=
f
2
,
since
the
commutativity
of
the
square
is
already
given.)
Again,
Remarks
5.1.2
apply.
We
call
(5.7)
a
pullback
square.
Another
name
for
pullback
is
fibred
prod-
uct.
This
name
is
partially
explained
by
the
following
fact:
when
Z
is
a
terminal
object
(and
s
and
t
are
the
only
maps
they
can
possibly
be),
a
pullback
of
the
diagram
(5.6)
is
simply
a
product
of
X
and
Y.
Examples
5.1.17
(Pullbacks
in
Set)
is
The
pullback
of
a
diagram
(5.6)
in
Set
P
=
{(x,
y)
∈
X
×
Y
|
s(x)
=
t(y)}
with
projections
p
1
and
p
2
given
by
p
1
(x,
y)
=
x
and
p
2
(x,
y)
=
y.
Although
you
might
not
be
familiar
with
general
pullbacks
in
Set,
there
are
at
least
two
instances
that
you
are
likely
to
have
met.
(a)
A
basic
construction
with
sets
and
functions
is
the
formation
of
inverse
images.
They
are
an
instance
of
pullbacks.
Indeed,
given
a
function
f
:
X
→
Y
and
a
subset
Y
0
⊆
Y,
we
obtain
a
new
set,
the
inverse
image
f
−1
Y
0
=
{x
∈
X
|
f
(x)
∈
Y
0
}
⊆
X,
and
a
new
function,
f
0
:
f
−1
Y
0
x
→
7
→
Y
0
f
(x).
116
Limits
We
also
have
the
inclusion
functions
j
:
Y
0
,→
Y
and
i
:
f
−1
Y
0
,→
X.
Putting
everything
together
gives
a
commutative
square
f
0
0
f
−1
Y
_
i
X
f
/
Y
0
_
/
Y.
(5.10)
j
The
data
we
started
with
was
the
lower-right
part
of
this
square
(X,
Y,
Y
0
,
f
and
j),
and
from
it
we
constructed
the
rest
of
the
square
(
f
−1
Y
0
,
f
0
and
i).
The
square
(5.10)
is
a
pullback.
Let
us
verify
this
in
detail.
Take
any
commutative
square
/
Y
0
h
A
g
X
/
Y.
f
_
j
We
must
show
that
there
is
a
unique
map
k
:
A
→
f
−1
Y
0
such
that
A
h
k
g
!
0
f
−1
Y
_
X
f
0
i
f
%
/
Y
0
_
/
Y
j
commutes.
For
uniqueness,
let
k
be
a
map
making
the
diagram
commute.
Then
for
all
a
∈
A,
we
have
i(k(a))
=
g(a),
that
is,
k(a)
=
g(a),
and
this
determines
k
uniquely.
For
existence,
first
note
that
for
all
a
∈
A
we
have
f
(g(a))
=
j(h(a))
∈
Y
0
,
so
g(a)
∈
f
−1
Y
0
.
Hence
we
may
define
k
:
A
→
f
−1
Y
0
by
k(a)
=
g(a)
for
all
a
∈
A.
Then
for
all
a
∈
A,
we
have
i(k(a))
=
k(a)
=
g(a)
and
f
0
(k(a))
=
f
(k(a))
=
f
(g(a))
=
j(h(a))
=
h(a).
Hence
i
◦
k
=
g
and
f
0
◦
k
=
h,
as
required.
5.1
Limits:
definition
and
examples
117
(b)
Intersection
of
subsets
provides
another
example
of
pullbacks.
Indeed,
let
X
and
Y
be
subsets
of
a
set
Z.
Then
/
Y
X
∩
Y
_
_
X
/
Z
is
a
pullback
square,
where
all
the
arrows
are
inclusions
of
subsets.
In
fact,
this
is
a
special
case
of
(a),
since
X
∩
Y
is
the
inverse
image
of
Y
⊆
Z
under
the
inclusion
map
X
,→
Z.
In
the
situation
of
Example
5.1.17(a),
where
we
have
a
map
f
:
X
→
Y
and
a
subset
Y
0
of
Y,
people
sometimes
say
that
f
−1
Y
0
is
obtained
by
‘pulling
Y
0
back’
along
f
:
hence
the
name.
The
definition
of
limit
We
have
now
looked
at
three
constructions:
products,
equalizers
and
pullbacks.
They
clearly
have
something
in
common.
Each
starts
with
some
objects
and
(in
the
case
of
equalizers
and
pullbacks)
some
maps
between
them.
In
each,
we
aim
to
construct
a
new
object
together
with
some
maps
from
it
to
the
original
objects,
with
a
universal
property.
Let
us
analyse
this
more
closely.
What
is
the
starting
data
in
each
construc-
tion?
For
(binary)
products,
it
is
a
pair
of
objects
X
Y.
(5.11)
/
/
Y.
(5.12)
For
equalizers,
it
is
a
diagram
X
s
t
For
pullbacks,
it
is
a
diagram
Y
t
X
s
/
Z.
(5.13)
In
Definition
4.1.25,
we
met
the
notion
of
generalized
element,
and
we
saw
there
that
the
‘figures’
in
a
geometric
object
can
often
be
described
by
maps
into
it.
For
instance,
a
curve
in
a
topological
space
A
can
be
thought
of
as
a
map
R
→
A.
Similarly,
an
object
of
a
category
A
amounts
to
a
functor
D
:
1
→
A
;
think
of
1
=
•
as
an
unlabelled
object
and
D
as
labelling
it
with
118
Limits
'
$
3
7
D(I)
f
I
p
I
A
∃!
f
¯
/
L
p
J
'
+
f
J
D(J)
D
&
%
Figure
5.1
The
definition
of
limit.
the
name
of
an
object
of
A
.
And
similarly
again,
a
map
in
a
category
A
is
a
functor
2
→
A
,
where
2
=
•
→
•
.
(Here
2
is
the
category
with
two
objects,
say
0
and
1,
with
one
map
0
→
1,
and
with
no
other
maps
except
for
identities.)
Finally,
if
we
take
I
to
be
one
of
the
categories
•
T
=
•
•
,
E
=
•
/
/
•
or
P
=
•
/
•
(5.14)
then
a
functor
I
→
A
consists
of
data
(5.11),
(5.12)
or
(5.13)
in
A
,
respec-
tively.
We
have
just
begun
to
use
the
convention
that
one
typeface
(A,
B,
C,
.
.
.
)
denotes
small
categories,
and
another
(A
,
B,
C
,
.
.
.
)
denotes
arbitrary
cate-
gories.
Although
not
strictly
necessary,
this
convention
is
helpful,
since
small
categories
and
arbitrary
categories
often
play
different
roles
in
the
theory.
Definition
5.1.18
Let
A
be
a
category
and
I
a
small
category.
A
functor
I
→
A
is
called
a
diagram
in
A
of
shape
I.
So
(5.11),
(5.12)
and
(5.13)
are
diagrams
of
shape
T,
E
and
P.
We
already
have
the
definitions
of
product
of
a
diagram
of
shape
T,
equalizer
of
a
diagram
of
shape
E,
and
pullback
of
a
diagram
of
shape
P.
We
now
unify
them
in
the
definition
of
limit
(Figure
5.1).
Definition
5.1.19
diagram
in
A
.
Let
A
be
a
category,
I
a
small
category,
and
D
:
I
→
A
a
(a)
A
cone
on
D
is
an
object
A
∈
A
(the
vertex
of
the
cone)
together
with
a
family
f
I
(5.15)
A
−→
D(I)
I∈I
5.1
Limits:
definition
and
examples
119
u
of
maps
in
A
such
that
for
all
maps
I
−→
J
in
I,
the
triangle
6
D(I)
f
I
A
f
J
(
Du
D(J)
commutes.
(Here
and
later,
we
abbreviate
D(u)
as
Du.)
p
I
(b)
A
limit
of
D
is
a
cone
L
−→
D(I)
with
the
property
that
for
any
I∈I
cone
(5.15)
on
D,
there
exists
a
unique
map
f
¯
:
A
→
L
such
that
p
I
◦
f
¯
=
f
I
for
all
I
∈
I.
The
maps
p
I
are
called
the
projections
of
the
limit.
Remarks
5.1.20
(a)
Loosely,
the
universal
property
says
that
for
any
A
∈
A
,
maps
A
→
L
correspond
one-to-one
with
cones
on
D
with
vertex
A.
p
I
g
(Any
map
g
:
A
→
L
gives
rise
to
a
cone
A
−→
D(I)
,
and
the
defini-
I∈I
tion
of
limit
is
that
for
each
A,
this
process
is
bijective.)
In
Section
6.1,
we
will
use
this
thought
to
rephrase
the
definition
of
limit
in
terms
of
repre-
sentability.
From
this
it
will
follow
that
limits
are
unique
up
to
canonical
isomorphism,
when
they
exist
(Corollary
6.1.2).
Alternatively,
uniqueness
can
be
proved
by
the
usual
kind
of
direct
argument,
as
in
Lemma
2.1.8.
p
I
(b)
If
L
−→
D(I)
is
a
limit
of
D,
we
sometimes
abuse
language
slightly
by
I∈I
referring
to
L
(rather
than
the
whole
cone)
as
the
limit
of
D.
For
emphasis,
p
I
we
sometimes
call
L
−→
D(I)
a
limit
cone.
We
write
L
=
lim
D.
I∈I
←I
Remark
(a)
can
then
be
stated
as:
A
map
into
lim
D
is
a
cone
on
D.
←I
(c)
By
assuming
from
the
outset
that
the
shape
category
I
is
small,
we
are
restricting
ourselves
to
what
are
officially
called
small
limits.
We
will
sel-
dom
be
interested
in
any
other
kind.
Examples
5.1.21
(Limit
shapes)
gories
T,
E
and
P
of
(5.14).
Let
A
be
any
category.
Recall
the
cate-
(a)
A
diagram
D
of
shape
T
in
A
is
a
pair
(X,
Y)
of
objects
of
A
.
A
cone
on
D
is
an
object
A
together
with
maps
f
1
:
A
→
X
and
f
2
:
A
→
Y
(as
in
Definition
5.1.1),
and
a
limit
of
D
is
a
product
of
X
and
Y.
More
generally,
let
I
be
a
set
and
write
I
for
the
discrete
category
on
I.
A
functor
D
:
I
→
A
is
an
I-indexed
family
(X
i
)
i∈I
of
objects
of
A
,
and
a
limit
of
D
is
exactly
a
product
of
the
family
(X
i
)
i∈I
.
In
particular,
a
limit
of
the
unique
functor
∅
→
A
is
a
terminal
object
of
A
,
where
∅
denotes
the
empty
category.
120
Limits
(b)
A
diagram
D
of
shape
E
in
A
is
a
parallel
pair
X
s
t
/
/
Y
of
maps
in
A
.
A
cone
on
D
consists
of
objects
and
maps
A
f
X
g
/
/
Y
s
t
such
that
s
◦
f
=
g
and
t
◦
f
=
g.
But
since
g
is
determined
by
f
,
it
is
equivalent
to
say
that
a
cone
on
D
consists
of
an
object
A
and
a
map
f
:
A
→
X
such
that
A
f
/
X
/
/
Y
s
t
is
a
fork.
A
limit
of
D
is
a
universal
fork
on
s
and
t,
that
is,
an
equalizer
of
s
and
t.
(c)
A
diagram
D
of
shape
P
in
A
consists
of
objects
and
maps
Y
X
s
/
Z
t
in
A
.
Performing
a
simplification
similar
to
that
in
(b),
we
see
that
a
cone
on
D
is
a
commutative
square
(5.8).
A
limit
of
D
is
a
pullback.
(d)
Let
I
=
(N,
≤)
op
.
A
diagram
D
:
I
→
A
consists
of
objects
and
maps
s
3
s
2
s
1
·
·
·
−→
X
2
−→
X
1
−→
X
0
.
For
example,
suppose
that
we
have
a
set
X
0
and
a
chain
of
subsets
·
·
·
⊆
X
2
⊆
X
1
⊆
X
0
.
The
inclusion
maps
form
a
diagram
in
Set
of
the
type
above,
and
its
limit
T
is
i∈N
X
i
.
In
this
and
similar
contexts,
limits
are
sometimes
referred
to
as
inverse
limits,
although
many
category
theorists
regard
this
usage
as
old-fashioned.
In
general,
the
limit
of
a
diagram
D
is
the
terminal
object
in
the
category
of
cones
on
D,
and
is
therefore
an
extremal
example
of
a
cone
on
D.
The
word
‘limit’
can
be
understood
as
meaning
‘on
the
boundary’,
rather
than
indicating
a
limiting
process
of
the
type
encountered
in
analysis.
Nevertheless,
the
two
ideas
make
contact
in
Example
5.1.21(d).
5.1
Limits:
definition
and
examples
121
We
have
said
little
so
far
about
which
limits
exist,
except
to
observe
in
Re-
mark
5.1.2(a)
that
they
do
not
exist
always.
We
now
show
that
in
many
familiar
categories,
all
limits
do
exist;
indeed,
we
can
construct
them
explicitly.
Example
5.1.22
Let
D
:
I
→
Set
and,
as
a
kind
of
thought
experiment,
let
us
ask
ourselves
what
lim
D
would
have
to
be
if
it
existed.
(We
do
not
know
yet
←I
that
it
does.)
We
would
have
lim
D
Set
1,
lim
D
←I
←I
{cones
on
D
with
vertex
1}
n
(x
I
)
I∈I
x
I
∈
D(I)
for
all
I
∈
I
and
(Du)(x
I
)
=
x
J
o
u
for
all
I
−→
J
in
I
,
(5.16)
where
the
second
isomorphism
is
by
Remark
5.1.20(a)
and
the
third
is
by
def-
inition
of
cone.
In
fact,
(5.16)
really
is
the
limit
of
D
in
Set,
with
projections
p
J
:
lim
D
→
D(J)
given
by
p
J
(x
I
)
I∈I
=
x
J
(Exercise
5.1.37).
So
in
Set,
all
←I
limits
exist.
Example
5.1.23
The
same
formula
gives
limits
in
categories
of
algebras
such
as
Grp,
Ring,
Vect
k
,
.
.
.
.
Of
course,
we
also
have
to
say
what
the
group/ring/.
.
.
structure
on
the
set
(5.16)
is,
but
this
works
in
the
most
straightforward
way
imaginable.
For
instance,
in
Vect
k
,
if
(x
I
)
I∈I
,
(y
I
)
I∈I
∈
lim
D
then
←I
(x
I
)
I∈I
+
(y
I
)
I∈I
=
(x
I
+
y
I
)
I∈I
.
Example
5.1.24
The
same
formula
also
gives
limits
in
Top.
The
topology
on
the
set
(5.16)
is
the
smallest
for
which
the
projection
maps
are
continuous.
Definition
5.1.25
(a)
Let
I
be
a
small
category.
A
category
A
has
limits
of
shape
I
if
for
every
diagram
D
of
shape
I
in
A
,
a
limit
of
D
exists.
(b)
A
category
has
all
limits
(or
properly,
has
small
limits)
if
it
has
limits
of
shape
I
for
all
small
categories
I.
Thus,
Set,
Top,
Grp,
Ring,
Vect
k
,
.
.
.
all
have
all
limits.
Similar
terminology
can
be
applied
to
special
classes
of
limits
(for
instance,
‘has
pullbacks’).
The
class
of
finite
limits
is
particularly
important.
By
defini-
tion,
a
category
is
finite
if
it
contains
only
finitely
many
maps
(in
which
case
it
also
contains
only
finitely
many
objects).
A
finite
limit
is
a
limit
of
shape
I
for
some
finite
category
I.
For
instance,
binary
products,
terminal
objects,
equalizers
and
pullbacks
are
all
finite
limits.
The
next
result
tells
us
that
all
limits
can
be
built
up
from
limits
of
just
a
few
familiar,
basic
types.
122
Limits
Proposition
5.1.26
Let
A
be
a
category.
(a)
If
A
has
all
products
and
equalizers
then
A
has
all
limits.
(b)
If
A
has
binary
products,
a
terminal
object
and
equalizers
then
A
has
finite
limits.
To
understand
the
idea,
consider
formula
(5.16)
for
limits
in
Set.
There,
Q
the
limit
of
a
diagram
D
is
described
as
the
subset
of
the
product
I∈I
D(I)
consisting
of
those
elements
for
which
certain
equations
hold.
We
saw
in
Ex-
ample
5.1.12
that
the
set
of
solutions
to
any
system
of
simultaneous
equations
can
be
described
via
products
and
equalizers.
Thus,
we
can
describe
any
limit
in
Set
in
terms
of
products
and
equalizers.
And
in
fact,
this
same
description
is
valid
in
any
category.
We
now
examine
this
idea
more
closely,
in
preparation
for
the
proof
(Ex-
ercise
5.1.38).
First-time
readers
may
wish
to
skip
the
next
two
paragraphs,
resuming
at
Example
5.1.27.
Equation
(5.16)
states
that
in
Set,
the
limit
of
a
diagram
D
:
I
→
Set
consists
Q
of
the
elements
(x
I
)
I∈I
∈
I∈I
D(I)
such
that
(Du)(x
J
)
=
x
K
u
in
D(K)
for
each
map
J
−→
K
in
I.
For
each
such
map
u,
define
maps
/
/
D(K)
s
u
t
u
Y
D(I)
I∈I
by
s
u
(x
I
)
I∈I
=
(Du)(x
J
),
t
u
(x
I
)
I∈I
=
x
K
.
Then
lim
D
is
the
set
of
families
x
=
(x
I
)
I∈I
satisfying
the
equation
s
u
(x)
=
t
u
(x)
←I
for
each
map
u
in
I.
It
follows
from
Example
5.1.12
that
lim
D
is
the
equalizer
←I
of
s
Y
I∈I
D(I)
t
/
/
Y
D(K)
u
J
−→K
in
I
where
s
and
t
are
the
maps
with
components
s
u
and
t
u
,
respectively.
We
have
now
described
any
limit
in
Set
in
terms
of
products
and
equal-
izers.
Although
our
argument
took
place
entirely
in
Set,
it
suggests
how
we
might
proceed
in
an
arbitrary
category.
With
this
in
mind,
the
proof
of
Propo-
sition
5.1.26
is
routine,
and
is
left
as
Exercise
5.1.38.
5.1
Limits:
definition
and
examples
123
Example
5.1.27
Let
CptHff
denote
the
category
of
compact
Hausdorff
spa-
ces
and
continuous
maps.
It
is
a
classic
exercise
in
topology
to
show
that
given
continuous
maps
s
and
t
from
a
topological
space
X
to
a
Hausdorff
space
Y,
the
subset
{x
∈
X
|
s(x)
=
t(x)}
of
X
is
closed.
From
this
it
follows
that
CptHff
has
equalizers.
Also,
Tychonoff’s
theorem
states
that
any
product
(in
Top)
of
compact
spaces
is
compact,
and
it
is
easy
to
show
that
any
product
(in
Top)
of
Hausdorff
spaces
is
Hausdorff.
From
this
it
follows
that
CptHff
has
all
products.
Hence
by
Proposition
5.1.26(a),
CptHff
has
all
limits.
Example
5.1.28
Recall
from
Example
5.1.15
that
kernels
provide
equalizers
in
Vect
k
.
By
Proposition
5.1.26(b),
finite
limits
in
Vect
k
can
always
be
ex-
pressed
in
terms
of
⊕
(binary
direct
sum),
{0},
and
kernels.
The
same
is
true
in
Ab.
Monics
For
functions
between
sets,
injectivity
is
an
important
concept.
For
maps
in
an
arbitrary
category,
injectivity
does
not
make
sense,
but
there
is
a
concept
that
plays
a
similar
role.
f
Let
A
be
a
category.
A
map
X
−→
Y
in
A
is
monic
(or
a
x
/
/
X
,
monomorphism)
if
for
all
objects
A
and
maps
A
Definition
5.1.29
x
0
f
◦
x
=
f
◦
x
0
=⇒
x
=
x
0
.
This
can
be
rephrased
suggestively
in
terms
of
generalized
elements:
f
is
monic
if
for
all
generalized
elements
x
and
x
0
of
X
(of
the
same
shape),
f
x
=
f
x
0
=⇒
x
=
x
0
.
Being
monic
is,
therefore,
the
generalized-element
analogue
of
injectivity.
Example
5.1.30
In
Set,
a
map
is
monic
if
and
only
if
it
is
injective.
Indeed,
if
f
is
injective
then
certainly
f
is
monic,
and
for
the
converse,
take
A
=
1.
Example
5.1.31
In
categories
of
algebras
such
as
Grp,
Vect
k
,
Ring,
etc.,
it
is
also
true
that
the
monic
maps
are
exactly
the
injections.
Again,
it
is
easy
to
show
that
injections
are
monic.
For
the
converse,
take
A
=
F(1)
where
F
is
the
free
functor
(Examples
2.1.3).
Why
is
the
definition
of
monic
in
a
chapter
on
limits?
Because
of
this:
124
Limits
f
Lemma
5.1.32
A
map
X
−→
Y
is
monic
if
and
only
if
the
square
1
X
1
X
f
/
X
f
/
Y
is
a
pullback.
Proof
Exercise
5.1.41.
The
significance
of
this
lemma
is
that
whenever
we
prove
a
result
about
limits,
a
result
about
monics
will
follow.
For
example,
we
will
soon
show
that
the
forgetful
functors
from
Grp,
Vect
k
,
etc.,
to
Set
preserve
limits
(in
a
sense
to
be
defined),
from
which
it
will
follow
immediately
that
they
also
preserve
monics.
This
in
turn
gives
an
alternative
proof
that
monics
in
these
categories
are
injective.
Exercises
5.1.33
Verify
that
in
the
category
of
vector
spaces,
the
product
of
two
vector
spaces
is
their
direct
sum
(Example
5.1.5).
5.1.34
Take
objects
and
maps
E
f
/
X
i
g
/
/
Y
in
some
category.
If
this
is
an
equalizer,
is
the
square
E
i
i
X
f
/
X
/
Y
g
necessarily
a
pullback?
What
about
the
converse?
Give
proofs
or
counterex-
amples.
5.1.35
Take
a
commutative
diagram
·
/
·
/
·
·
/
·
/
·
in
some
category.
Suppose
that
the
right-hand
square
is
a
pullback.
Show
that
the
left-hand
square
is
a
pullback
if
and
only
if
the
outer
rectangle
is
a
pullback.
5.1
Limits:
definition
and
examples
125
p
I
5.1.36
Let
D
:
I
→
A
be
a
diagram
and
L
−→
D(I)
a
limit
cone
on
D.
I∈I
h
(a)
Prove
that
whenever
A
h
0
/
/
L
are
maps
such
that
p
I
◦
h
=
p
I
◦
h
0
for
all
I
∈
I,
then
h
=
h
0
.
(b)
What
does
the
result
of
(a)
mean
when
I
is
the
two-object
discrete
cate-
gory,
A
=
Set,
and
A
=
1?
Answer
without
using
any
category-theoretic
terminology.
5.1.37
Show
that
the
set
(5.16)
in
Example
5.1.22
really
is
a
limit
of
D.
5.1.38
In
this
exercise,
you
will
prove
Proposition
5.1.26,
following
the
plan
described
after
the
statement
of
that
proposition.
(a)
Let
A
be
a
category
with
all
products
and
equalizers.
Let
D
:
I
→
A
be
a
diagram
in
A
.
Define
maps
s
Y
I∈I
D(I)
t
/
/
Y
D(K)
u
J
−→K
in
I
u
as
follows:
given
J
−→
K
in
I,
the
u-component
of
s
is
the
composite
Y
pr
J
Du
D(I)
−→
D(J)
−→
D(K)
I∈I
(where
pr
denotes
a
product
projection),
and
the
u-component
of
t
is
pr
K
.
p
Q
Let
L
−→
I∈I
D(I)
be
the
equalizer
of
s
and
t,
and
write
p
I
for
the
I-
p
I
component
of
p.
Show
that
L
−→
D(I)
is
a
limit
cone
on
D,
thus
I∈I
proving
Proposition
5.1.26(a).
(b)
Adapt
the
argument
to
prove
Proposition
5.1.26(b).
5.1.39
Prove
that
a
category
with
pullbacks
and
a
terminal
object
has
all
finite
limits.
5.1.40
Let
A
be
a
category
and
A
∈
A
.
A
subobject
of
A
is
an
isomorphism
class
of
monics
into
A.
More
precisely,
let
Monic(A)
be
the
full
subcategory
of
A
/A
whose
objects
are
the
monics;
then
a
subobject
of
A
is
an
isomorphism
class
of
objects
of
Monic(A).
m
m
0
(a)
Let
X
−→
A
and
X
0
−→
A
be
monics
in
Set.
Show
that
m
and
m
0
are
isomorphic
in
Monic(A)
if
and
only
if
they
have
the
same
image.
Deduce
that
the
subobjects
of
A
are
in
canonical
one-to-one
correspondence
with
the
subsets
of
A.
126
Limits
(b)
Part
(a)
says
that
in
Set,
subobjects
are
subsets.
What
are
subobjects
in
Grp,
Ring
and
Vect
k
?
(c)
What
are
subobjects
in
Top?
(Careful!)
5.1.41
Prove
Lemma
5.1.32.
5.1.42
Let
X
0
f
0
m
0
A
0
/
X
m
f
/
A
be
a
pullback
square
in
some
category.
Show
that
if
m
is
monic
then
so
is
m
0
.
(We
already
know
this
in
the
category
of
sets,
by
Example
5.1.17(a).)
5.2
Colimits:
definition
and
examples
We
have
seen
that
examples
of
limits
occur
throughout
mathematics.
It
there-
fore
makes
sense
to
examine
the
dual
concept,
colimit,
and
ask
whether
it
is
similarly
ubiquitous.
By
dualizing,
we
can
write
down
the
definition
of
colimit
immediately.
We
then
specialize
to
sums,
coequalizers
and
pushouts,
the
duals
of
products,
equalizers
and
pullbacks.
There
are
two
common
conventions
for
naming
dual
concepts:
sometimes
we
add
or
subtract
the
prefix
‘co’
(as
in
limit/colimit),
and
sometimes
we
use
‘left’
and
‘right’
(as
for
adjoints).
There
are
also
some
irregular
names,
such
as
terminal/initial
object
and
pullback/pushout.
Definition
5.2.1
Let
A
be
a
category
and
I
a
small
category.
Let
D
:
I
→
A
be
a
diagram
in
A
,
and
write
D
op
for
the
corresponding
functor
I
op
→
A
op
.
A
cocone
on
D
is
a
cone
on
D
op
,
and
a
colimit
of
D
is
a
limit
of
D
op
.
Explicitly,
a
cocone
on
D
is
an
object
A
∈
A
(the
vertex
of
the
cocone)
together
with
a
family
f
I
D(I)
−→
A
I∈I
(5.17)
5.2
Colimits:
definition
and
examples
127
u
of
maps
in
A
such
that
for
all
maps
I
−→
J
in
I,
the
diagram
D(I)
f
I
(
6
A
Du
D(J)
f
J
commutes.
A
colimit
of
D
is
a
cocone
p
I
D(I)
−→
C
I∈I
with
the
property
that
for
any
cocone
(5.17)
on
D,
there
is
a
unique
map
f
¯
:
C
→
A
such
that
f
¯
◦
p
I
=
f
I
for
all
I
∈
I.
The
associated
picture
is
the
mirror
image
of
Figure
5.1.
Of
course,
Remarks
5.1.20
apply
equally
here.
We
write
(the
vertex
of)
the
colimit
as
lim
D,
and
call
the
maps
p
I
coprojections.
→I
Sums
Definition
5.2.2
A
sum
or
coproduct
is
a
colimit
over
a
discrete
category.
(That
is,
it
is
a
colimit
of
shape
I
for
some
discrete
category
I.)
Let
(X
i
)
i∈I
be
a
family
of
objects
of
a
category.
Their
sum
(if
it
exists)
is
written
P
`
P
as
i∈I
X
i
or
i∈I
X
i
.
When
I
is
a
finite
set
{1,
.
.
.
,
n},
we
write
i∈I
X
i
as
X
1
+
·
·
·
+
X
n
,
or
as
0
if
n
=
0.
Example
5.2.3
By
the
dual
of
Example
5.1.9,
a
sum
of
the
empty
family
is
exactly
an
initial
object.
Example
5.2.4
Sums
in
Set
were
described
in
Section
3.1.
Let
us
look
in
detail
at
the
universal
property,
in
the
case
of
binary
sums.
Take
two
sets,
X
1
and
X
2
.
Form
their
sum,
X
1
+
X
2
,
and
consider
the
inclusions
X
1
p
1
/
X
1
+
X
2
o
p
2
X
2
.
This
is
a
colimit
cocone.
To
prove
this,
we
have
to
prove
the
following
universal
property:
for
any
diagram
X
1
f
1
/
A
o
f
2
X
2
128
Limits
of
sets
and
functions,
there
is
a
unique
function
f
¯
:
X
1
+
X
2
→
A
making
X
1
f
1
p
1
p
2
%
X
9
1
+
X
2
f
¯
/
*
4
A
f
2
X
2
commute.
Now,
we
noted
in
Section
3.1
that
p
1
and
p
2
are
injections
whose
images
partition
X
1
+
X
2
.
This
means
that
every
element
x
of
X
1
+
X
2
is
either
equal
to
p
1
(x
1
)
for
some
x
1
∈
X
1
(and
this
x
1
is
then
unique),
or
equal
to
p
2
(x
2
)
for
some
x
2
∈
X
2
(and
this
x
2
is
then
unique),
but
not
both.
So
we
may
define
f
¯
(x)
to
be
equal
to
f
1
(x
1
)
in
the
first
case
and
f
2
(x
2
)
in
the
second.
This
defines
a
function
f
¯
making
the
diagram
commute,
and
it
is
clearly
the
unique
function
that
does
so.
Example
5.2.5
Let
X
1
and
X
2
be
vector
spaces.
There
are
linear
maps
X
1
i
1
/
X
1
⊕
X
2
o
i
2
X
2
(5.18)
defined
by
i
1
(x
1
)
=
(x
1
,
0)
and
i
2
(x
2
)
=
(0,
x
2
),
and
it
can
be
checked
that
(5.18)
is
a
colimit
cocone
in
Vect
k
.
Hence
binary
direct
sums
are
sums
in
the
categor-
ical
sense.
This
is
remarkable,
since
we
saw
in
Example
5.1.5
that
X
1
⊕
X
2
is
also
the
product
of
X
1
and
X
2
!
Contrast
this
with
the
category
of
sets
(or
almost
any
other
category),
where
sums
and
products
are
very
different.
Example
5.2.6
Let
(A,
≤)
be
an
ordered
set.
Upper
bounds
and
least
up-
per
bounds
(or
joins)
in
A
are
defined
by
dualizing
the
definitions
in
Exam-
ple
5.1.6,
and,
dually,
they
are
sums
in
the
corresponding
category.
The
join
of
W
a
family
(x
i
)
i∈I
is
written
as
i∈I
x
i
.
In
the
binary
case
(where
I
has
two
ele-
ments),
the
join
of
x
1
and
x
2
is
written
as
x
1
∨
x
2
.
A
join
of
the
empty
family
(where
I
=
∅)
is
an
initial
object
of
the
category
A,
as
in
Example
5.2.3.
Equiv-
alently,
it
is
a
least
element
of
A:
an
element
0
∈
A
such
that
0
≤
a
for
all
a
∈
A.
For
instance,
in
(R,
≤),
join
is
supremum
and
there
is
no
least
element.
In
a
power
set
(P(S
),
⊆),
join
is
union
and
the
least
element
is
∅.
In
(N,
|),
join
is
lowest
common
multiple
and
the
least
element
is
1
(since
1
divides
everything).
So
in
this
order
on
the
natural
numbers,
1
is
least;
but
also,
everything
divides
0,
so
0
is
greatest!
5.2
Colimits:
definition
and
examples
129
Coequalizers
We
continue
to
write
E
for
the
category
•
⇒
•
.
Definition
5.2.7
A
coequalizer
is
a
colimit
of
shape
E.
In
other
words,
given
a
diagram
X
s
t
/
/
Y
,
a
coequalizer
of
s
and
t
is
a
map
p
Y
−→
C
satisfying
p
◦
s
=
p
◦
t
and
universal
with
this
property.
We
will
see
that
coequalizers
are
something
like
quotients.
But
first,
we
need
some
background
material
on
equivalence
relations.
Remarks
5.2.8
A
binary
relation
R
on
a
set
A
can
be
viewed
as
a
subset
R
⊆
A
×
A.
Think
of
(a,
a
0
)
∈
R
as
meaning
‘a
and
a
0
are
related’.
We
can
speak
of
one
relation
S
on
A
‘containing’
another
such
relation,
R.
This
means
that
R
⊆
S
:
whenever
a
and
a
0
are
R-related,
they
are
also
S
-related.
We
will
need
to
use
the
fact
that
for
any
binary
relation
R
on
a
set
A,
there
is
a
smallest
equivalence
relation
∼
containing
R.
This
is
called
the
equiva-
lence
relation
generated
by
R.
‘Smallest’
means
that
any
equivalence
relation
containing
R
also
contains
∼.
We
can
construct
∼
as
the
intersection
of
all
equivalence
relations
on
A
con-
taining
R,
since
the
intersection
of
any
family
of
equivalence
relations
is
again
an
equivalence
relation.
There
is
also
an
explicit
construction.
The
rough
idea
is
as
follows:
writing
x
→
y
to
mean
(x,
y)
∈
R,
we
should
have
a
∼
a
0
if
and
only
if
there
is
a
zigzag
such
as
a
→
b
←
c
←
d
→
e
←
a
0
between
a
and
a
0
.
To
make
this
precise,
we
first
define
a
relation
S
on
A
by
S
=
{(a,
a
0
)
∈
A
×
A
|
(a,
a
0
)
∈
R
or
(a
0
,
a)
∈
R}
(which
enlarges
R
to
a
symmetric
relation),
then
define
∼
by
declaring
that
a
∼
a
0
if
and
only
if
there
exist
n
≥
0
and
a
0
,
.
.
.
,
a
n
∈
A
such
that
a
=
a
0
,
(a
0
,
a
1
)
∈
S
,
(a
1
,
a
2
)
∈
S
,
.
.
.
,
(a
n−1
,
a
n
)
∈
S
,
a
n
=
a
0
(which
forces
reflexivity
and
transitivity,
while
preserving
the
symmetry).
Next,
recall
some
facts
about
equivalence
relations
from
Section
3.1.
Given
any
equivalence
relation
∼
on
a
set
A,
we
can
construct
the
set
A/∼
of
equiv-
alence
classes
and
the
quotient
map
p
:
A
→
A/∼.
This
quotient
map
p
is
surjective
and
has
the
property
that
p(a)
=
p(a
0
)
⇐⇒
a
∼
a
0
,
for
a,
a
0
∈
A.
We
saw
that
for
any
set
B,
the
maps
A/∼
→
B
correspond
one-to-one
(via
composition
with
p)
with
the
maps
f
:
A
→
B
such
that
∀a,
a
0
∈
A,
a
∼
a
0
=⇒
f
(a)
=
f
(a
0
).
(5.19)
130
Limits
Finally,
let
us
consider
this
universal
property
in
the
case
where
∼
is
the
equivalence
relation
generated
by
some
relation
R.
Condition
(5.19)
is
then
equivalent
to:
∀a,
a
0
∈
A,
(a,
a
0
)
∈
R
=⇒
f
(a)
=
f
(a
0
).
(5.20)
(Proof:
define
an
equivalence
relation
≈
on
A
by
a
≈
a
0
⇐⇒
f
(a)
=
f
(a
0
).
Condition
(5.19)
says
that
∼
⊆
≈,
and
condition
(5.20)
that
R
⊆
≈.
But
∼
is
the
smallest
equivalence
relation
containing
R,
so
these
statements
are
equivalent.)
In
conclusion,
for
any
set
B,
the
maps
A/∼
→
B
correspond
one-to-one
with
the
maps
f
:
A
→
B
satisfying
(5.20).
Example
5.2.9
Take
sets
and
functions
X
s
t
/
/
Y
.
To
find
the
coequalizer
of
s
and
t,
we
must
construct
in
some
canonical
way
a
set
C
and
a
function
p
:
Y
→
C
such
that
p(s(x))
=
p(t(x))
for
all
x
∈
X.
So,
let
∼
be
the
equivalence
relation
on
Y
generated
by
s(x)
∼
t(x)
for
all
x
∈
X.
(In
other
words,
∼
is
generated
by
the
relation
R
=
{(s(x),
t(x))
|
x
∈
X}
on
Y.)
Take
the
quotient
map
p
:
Y
→
Y/∼.
By
the
correspondence
described
in
Remarks
5.2.8,
this
is
indeed
the
coequalizer
of
s
and
t.
Example
5.2.10
For
each
pair
of
homomorphisms
A
s
t
/
/
B
in
Ab,
there
is
a
homomorphism
t
−
s
:
A
→
B,
which
gives
rise
to
a
subgroup
im(t
−
s)
of
B.
The
coequalizer
of
s
and
t
is
the
canonical
homomorphism
B
→
B/im(t
−
s).
(Compare
Example
5.1.15.)
Pushouts
Definition
5.2.11
A
pushout
is
a
colimit
of
shape
/
•
•
P
op
=
.
•
In
other
words,
the
pushout
of
a
diagram
X
t
Z
s
/
Y
(5.21)
5.2
Colimits:
definition
and
examples
131
is
(if
it
exists)
a
commutative
square
X
t
s
Z
/
Y
/
·
that
is
universal
as
such.
In
other
words
still,
a
pushout
in
a
category
A
is
a
pullback
in
A
op
.
Example
5.2.12
Take
a
diagram
(5.21)
in
Set.
Its
pushout
P
is
(Y
+
Z)/∼,
where
∼
is
the
equivalence
relation
on
Y
+
Z
generated
by
s(x)
∼
t(x)
for
all
x
∈
X.
The
coprojection
Y
→
P
sends
y
∈
Y
to
its
equivalence
class
in
P,
and
similarly
for
the
coprojection
Z
→
P.
For
example,
let
Y
and
Z
be
subsets
of
some
set
A.
Then
/
Y
Y
∩
Z
_
_
Z
/
Y
∪
Z
is
a
pushout
square
in
Set.
(It
is
also
a
pullback
square!
This
coincidence
is
a
special
property
of
the
category
of
sets.)
You
can
check
this
by
verifying
the
universal
property
or
by
using
the
formula
just
stated.
In
this
case,
the
formula
takes
the
two
sets
Y
and
Z,
places
them
side
by
side
(giving
Y
+
Z),
then
glues
the
subset
Y
∩
Z
of
Y
to
the
subset
Y
∩
Z
of
Z
(giving
(Y
+
Z)/∼
=
Y
∪
Z).
Example
5.2.13
If
A
is
a
category
with
an
initial
object
0,
and
if
Y,
Z
∈
A
,
then
a
pushout
of
the
unique
diagram
/
Y
0
Z
is
exactly
a
sum
of
Y
and
Z.
Example
5.2.14
The
van
Kampen
theorem
(Example
0.9)
says
that
given
a
pushout
square
in
Top
satisfying
certain
further
hypotheses,
the
square
in
Grp
obtained
by
taking
fundamental
groups
throughout
is
also
a
pushout.
Here
is
one
more
shape
of
colimit,
dual
to
that
in
Example
5.1.21(d).
Example
5.2.15
A
diagram
D
:
(N,
≤)
→
A
consists
of
objects
and
maps
s
1
s
2
s
3
X
0
−→
X
1
−→
X
2
−→
·
·
·
132
Limits
z
D
y
x
D
(b)
(a)
Figure
5.2
Sphere
as
(a)
a
limit,
and
(b)
a
colimit.
in
A
.
Colimits
of
such
diagrams
are
traditionally
called
direct
limits.
Al-
though
the
old
terms
‘inverse
limit’
(Example
5.1.21(d))
and
‘direct
limit’
are
made
redundant
by
the
general
categorical
terms
‘limit’
and
‘colimit’
respec-
tively,
it
is
worth
being
aware
of
them.
With
all
these
examples
in
mind,
we
now
write
down
a
general
formula
for
colimits
in
Set.
Example
5.2.16
The
colimit
of
a
diagram
D
:
I
→
Set
is
given
by
!,
X
lim
D
=
D(I)
∼
→I
I∈I
where
∼
is
the
equivalence
relation
on
P
D(I)
generated
by
x
∼
(Du)(x)
u
for
all
I
−→
J
in
I
and
x
∈
D(I).
To
see
this,
note
that
for
any
set
A,
the
maps
X
D(I)
∼
→
A
correspond
bijectively
with
the
maps
f
:
P
D(I)
→
A
such
that
f
(x)
=
f
(Du)(x)
for
all
u
and
x
(by
Remarks
5.2.8).
These
in
turn
correspond
to
families
of
f
I
such
that
f
I
(x)
=
f
J
(Du)(x)
for
all
u
and
x;
but
these
maps
D(I)
−→
A
I∈I
are
exactly
the
cocones
on
D
with
vertex
A.
5.2
Colimits:
definition
and
examples
133
There
is
a
kind
of
duality
between
the
formulas
for
limits
in
Set
(Exam-
ple
5.1.22)
and
colimits
in
Set.
Whereas
the
limit
is
constructed
as
a
subset
of
a
product,
the
colimit
is
a
quotient
of
a
sum.
Figure
5.2
is
intended
to
convey
the
difference
in
flavour
between
limits
and
colimits,
in
a
particular
topological
context.
In
elementary
texts,
surfaces
are
almost
always
seen
as
subsets
of
Euclidean
space
R
3
,
with
the
sphere
S
2
typically
defined
as
(x,
y,
z)
∈
R
3
|
x
2
+
y
2
+
z
2
=
1
.
This
is
a
subspace
of
the
product
space
R
3
=
R
×
R
×
R,
which
suggests
that
it
is
a
limit.
Indeed,
the
sphere
is
the
equalizer
S
2
/
R
3
s
t
/
/
R
where
the
maps
s,
t
:
R
3
→
R
are
given
by
s(x,
y,
z)
=
x
2
+
y
2
+
z
2
,
t(x,
y,
z)
=
1.
(An
equation
is
captured
by
an
equalizer.)
In
more
advanced
mathematics,
however,
this
point
of
view
is
used
less
of-
ten.
A
surface
can
instead
be
thought
of
as
the
gluing-together
of
lots
of
little
patches,
each
isomorphic
to
the
open
unit
disk
D.
For
example,
we
could
in
principle
construct
an
entire
bicycle
inner
tube
by
gluing
together
a
large
num-
ber
of
puncture-repair
patches.
Figure
5.2(b)
shows
the
simpler
example
of
a
sphere
made
up
of
two
disks
glued
together.
This
realizes
the
sphere
as
a
quotient
(gluing)
of
the
sum
(disjoint
union)
of
the
two
copies
of
D,
suggest-
ing
that
we
have
constructed
the
sphere
as
a
colimit.
Indeed,
the
sphere
is
the
coequalizer
/
/
S
2
S
1
×
(0,
1)
/
D
+
D
where
S
1
is
the
circle,
the
cylinder
S
1
×
(0,
1)
is
the
intersection
of
the
two
copies
of
D
(the
central
belt
of
Figure
5.2(b)),
and
the
two
maps
into
D
+
D
are
the
inclusions
of
the
cylinder
into
the
first
and
second
copies
of
D.
One
disadvantage
of
the
limit
point
of
view
is
that
it
makes
an
arbitrary
choice
of
coordinate
system.
It
is
generally
best
to
think
of
spaces
as
free-
standing
objects,
existing
independently
of
any
particular
embedding
into
Eu-
clidean
space.
One
disadvantage
of
the
colimit
point
of
view
is
that
it
makes
an
arbitrary
choice
of
decomposition.
For
example,
we
could
decompose
the
sphere
into
three
patches
rather
than
two,
or
use
a
different
two
patches
from
those
shown.
134
Limits
The
colimit
point
of
view
has
the
upper
hand
in
modern
geometry.
(If
you
are
familiar
with
the
definition
of
manifold,
you
will
recognize
that
an
atlas
is
essentially
a
way
of
viewing
a
manifold
as
a
colimit
of
Euclidean
balls.)
One
reason
for
this
is
that
we
are
often
concerned
with
maps
out
of
spaces
X,
such
as
maps
X
→
R.
Maps
out
of
a
colimit
are
easy;
it
is
in
the
very
definition
of
colimit
that
we
know
what
the
maps
out
of
it
are.
Epics
f
Let
A
be
a
category.
A
map
X
−→
Y
in
A
is
epic
(or
an
g
/
/
Z
,
epimorphism)
if
for
all
objects
Z
and
maps
Y
Definition
5.2.17
g
0
g
◦
f
=
g
0
◦
f
=⇒
g
=
g
0
.
This
is
the
formal
dual
of
the
definition
of
monic.
(In
other
words,
an
epic
in
A
is
a
monic
in
A
op
.)
It
is
in
some
sense
the
categorical
version
of
surjec-
tivity.
But
whereas
the
definition
of
monic
closely
resembles
the
definition
of
injective,
the
definition
of
epic
does
not
look
much
like
the
definition
of
sur-
jective.
The
following
examples
confirm
that
in
categories
where
surjectivity
makes
sense,
it
is
only
sometimes
equivalent
to
being
epic.
Example
5.2.18
In
Set,
a
map
is
epic
if
and
only
if
it
is
surjective.
If
f
is
surjective
then
certainly
f
is
epic.
To
see
the
converse,
take
Z
to
be
a
two-
element
set
{true,
false},
take
g
to
be
the
characteristic
function
of
the
image
of
f
(as
defined
in
Section
3.1),
and
take
g
0
to
be
the
function
with
constant
value
true.
Any
isomorphism
in
any
category
is
both
monic
and
epic.
In
Set,
the
con-
verse
also
holds,
since
any
injective
surjective
function
is
invertible
(Exam-
ple
1.1.5).
Example
5.2.19
In
categories
of
algebras,
any
surjective
map
is
certainly
epic.
In
some
such
categories,
including
Ab,
Vect
k
and
Grp,
the
converse
also
holds.
(The
proof
is
straightforward
for
Ab
and
Vect
k
,
but
much
harder
for
Grp.)
However,
there
are
other
categories
of
algebras
where
it
fails.
For
instance,
in
Ring,
the
inclusion
Z
,→
Q
is
epic
but
not
surjective
(Exer-
cise
5.2.23).
This
is
also
an
example
of
a
map
that
is
monic
and
epic
but
not
an
isomorphism.
Example
5.2.20
In
the
category
of
Hausdorff
topological
spaces
and
contin-
uous
maps,
any
map
with
dense
image
is
epic.
5.2
Colimits:
definition
and
examples
135
Of
course,
there
is
a
dual
of
Lemma
5.1.32,
saying
that
a
map
is
epic
if
and
only
if
a
certain
square
is
a
pushout.
Exercises
5.2.21
Let
X
s
t
/
/
Y
be
maps
in
some
category.
Prove
that
s
=
t
if
and
only
if
the
equalizer
of
s
and
t
exists
and
is
an
isomorphism,
if
and
only
if
the
coequalizer
of
s
and
t
exists
and
is
an
isomorphism.
5.2.22
(a)
Let
X
be
a
set
and
f
:
X
→
X
a
map.
Describe
the
coequalizer
of
f
X
1
/
/
X
in
Set
as
explicitly
as
possible.
(b)
Do
the
same
in
Top
rather
than
Set.
When
X
is
the
circle
S
1
,
find
an
f
such
that
the
coequalizer
is
an
uncountable
space
with
the
indiscrete
topology.
5.2.23
(a)
Prove
that
in
the
category
of
monoids,
the
inclusion
(N,
+,
0)
,→
(Z,
+,
0)
is
epic,
even
though
it
is
not
surjective.
(b)
Prove
that
in
the
category
of
rings,
the
inclusion
Z
,→
Q
is
epic,
even
though
it
is
not
surjective.
5.2.24
(Compare
Exercise
5.1.40.)
Let
A
be
a
category
and
A
∈
A
.
Define
a
quotient
object
of
A
to
be
an
isomorphism
class
of
epics
out
of
A.
That
is,
let
Epic(A)
be
the
full
subcategory
of
A/A
whose
objects
are
the
epics;
then
a
quotient
object
of
A
is
an
isomorphism
class
of
objects
of
Epic(A).
e
e
0
(a)
Let
A
−→
X
and
A
−→
X
0
be
epics
in
Set.
Show
that
e
and
e
0
are
isomor-
phic
in
Epic(A)
if
and
only
if
they
induce
the
same
equivalence
relation
on
A.
Deduce
that
the
quotient
objects
of
A
are
in
canonical
one-to-one
correspondence
with
the
equivalence
relations
on
A.
(b)
Assuming
the
(nontrivial)
fact
that
the
epics
in
Grp
are
the
surjections,
show
that
the
quotient
objects
of
a
group
correspond
one-to-one
with
its
normal
subgroups.
(The
name
‘quotient
object’
is
not
standard,
and
indeed
there
is
no
standard
name
for
it.
Arguably,
‘quotient
object’
would
be
more
suitable
for
an
isomor-
phism
class
of
regular
epics,
as
defined
in
the
following
exercises.)
5.2.25
A
map
m
:
A
→
B
is
regular
monic
if
there
exist
an
object
C
and
maps
B
⇒
C
of
which
m
is
an
equalizer.
A
map
m
:
A
→
B
is
split
monic
if
there
exists
a
map
e
:
B
→
A
such
that
em
=
1
A
.
(a)
Show
that
split
monic
=⇒
regular
monic
=⇒
monic.
136
Limits
(b)
In
Ab,
show
that
all
monics
are
regular
but
not
all
monics
are
split.
(Hint
for
the
first
part:
equalizers
in
Ab
are
calculated
as
in
Example
5.1.15.)
(c)
In
Top,
describe
the
regular
monics,
and
find
a
monic
that
is
not
regular.
5.2.26
Dualizing
the
definitions
in
Exercise
5.2.25
gives
definitions
of
regu-
lar
and
split
epic.
(a)
We
saw
in
Example
5.2.19
that
a
map
may
be
monic
and
epic
but
not
an
isomorphism.
Prove
that
in
any
category,
a
map
is
an
isomorphism
if
and
only
if
it
is
both
monic
and
regular
epic.
(b)
Using
the
assumption
that
our
category
of
sets
satisfies
the
axiom
of
choice
(Section
3.1),
show
that
epic
⇐⇒
regular
epic
⇐⇒
split
epic
in
Set.
(c)
Let
us
say
that
a
category
A
satisfies
the
axiom
of
choice
if
all
epics
in
A
are
split.
Prove
that
neither
Top
nor
Grp
satisfies
the
axiom
of
choice.
5.2.27
The
result
of
Exercise
5.1.42
can
be
phrased
as
‘the
class
of
monics
is
stable
under
pullback’.
It
is
also
a
fact
that
the
composite
of
two
monics
is
always
monic;
we
say
that
the
class
of
monics
is
‘closed
under
composition’.
Consider
the
following
six
classes
of
map:
monics,
regular
monics,
split
monics,
epics,
regular
epics,
split
epics.
Determine
whether
each
class
is
stable
under
pullback
or
closed
under
compo-
sition.
5.3
Interactions
between
functors
and
limits
We
saw
in
Example
5.1.23
that
limits
in
categories
such
as
Grp,
Ring
and
Vect
k
can
be
computed
by
first
taking
the
limit
in
the
category
of
sets,
then
equipping
the
result
with
a
suitable
algebraic
structure.
On
the
other
hand,
col-
imits
in
these
categories
are
unlike
colimits
in
Set.
For
example,
the
underlying
set
of
the
initial
object
of
Grp
(which
has
one
element)
is
not
the
initial
object
of
Set
(which
has
no
elements),
and
the
underlying
set
of
the
direct
sum
X
⊕
Y
of
two
vector
spaces
is
not
the
sum
of
the
underlying
sets
of
X
and
Y.
So,
these
forgetful
functors
interact
well
with
limits
and
badly
with
colimits.
In
this
section,
we
develop
terminology
that
will
enable
us
to
express
these
thoughts
precisely.
5.3
Interactions
between
functors
and
limits
137
Definition
5.3.1
(a)
Let
I
be
a
small
category.
A
functor
F
:
A
→
B
pre-
serves
limits
of
shape
I
if
for
all
diagrams
D
:
I
→
A
and
all
cones
p
I
A
−→
D(I)
on
D,
I∈I
p
I
is
a
limit
cone
on
D
in
A
F
p
I
=⇒
F(A)
−→
FD(I)
is
a
limit
cone
on
F
◦
D
in
B.
A
−→
D(I)
I∈I
I∈I
(b)
A
functor
F
:
A
→
B
preserves
limits
if
it
preserves
limits
of
shape
I
for
all
small
categories
I.
(c)
Reflection
of
limits
is
defined
as
in
(a),
but
with
⇐=
in
place
of
=⇒.
Of
course,
the
same
terminology
applies
to
colimits.
Here
is
a
different
way
to
state
the
definition
of
preservation.
A
functor
F
:
A
→
B
preserves
limits
if
and
only
if
it
has
the
following
property:
whenever
D
:
I
→
A
is
a
diagram
that
has
a
limit,
the
composite
F
◦
D
:
I
→
B
also
has
a
limit,
and
the
canonical
map
F
lim
D
→
lim
(F
◦
D)
←I
←I
is
an
isomorphism.
Here
the
‘canonical
map’
has
I-component
F(p
)
I
F
lim
D
−→
F(D(I)),
←I
where
p
I
is
the
Ith
projection
of
the
limit
cone
on
D.
In
particular,
if
F
preserves
limits
then
F
lim
D
lim
(F
◦
D)
←I
←I
(5.22)
whenever
D
is
a
diagram
with
a
limit.
Preservation
of
limits
says
more
than
(5.22)
does:
the
left-
and
right-hand
sides
are
required
to
be
not
just
isomor-
phic,
but
isomorphic
in
a
particular
way.
Nevertheless,
we
will
sometimes
omit
this
check,
acting
as
if
preservation
means
only
that
(5.22)
holds.
Example
5.3.2
The
forgetful
functor
U
:
Top
→
Set
preserves
both
limits
and
colimits.
(As
we
will
see,
this
follows
from
the
fact
that
U
has
adjoints
on
both
sides.)
It
does
not
reflect
all
limits
or
all
colimits.
For
instance,
choose
any
non-discrete
spaces
X
and
Y,
and
let
Z
be
the
set
U(X)
×
U(Y)
equipped
with
the
discrete
topology.
(All
that
matters
here
is
that
the
topology
on
Z
is
strictly
larger
than
the
product
topology.)
Then
we
have
a
cone
X
←
Z
→
Y
(5.23)
138
Limits
in
Top
whose
image
in
Set
is
the
product
cone
U(X)
←
U(X)
×
U(Y)
→
U(Y).
But
(5.23)
is
not
a
product
cone
in
Top,
since
the
discrete
topology
on
U(X)
×
U(Y)
is
not
the
product
topology.
Example
5.3.3
In
the
first
paragraph
of
this
section,
we
observed
that
the
forgetful
functor
Grp
→
Set
does
not
preserve
initial
objects
and
that
the
for-
getful
functor
Vect
k
→
Set
does
not
preserve
binary
sums.
Forgetful
functors
out
of
categories
of
algebras
very
seldom
preserve
all
colimits.
Example
5.3.4
We
also
saw
that
(in
the
examples
mentioned)
forgetful
func-
tors
on
categories
of
algebras
do
preserve
limits.
In
fact,
something
stronger
is
true.
Let
us
examine
the
case
of
binary
products
in
Grp,
although
all
of
the
following
can
be
said
for
any
limits
in
any
of
the
categories
Grp,
Ab,
Vect
k
,
Ring,
etc.
Take
groups
X
1
and
X
2
.
We
can
form
the
product
set
U(X
1
)
×
U(X
2
),
which
comes
equipped
with
projections
p
1
p
2
U(X
1
)
←−
U(X
1
)
×
U(X
2
)
−→
U(X
2
).
I
claim
that
there
is
exactly
one
group
structure
on
the
set
U(X
1
)
×
U(X
2
)
with
the
property
that
p
1
and
p
2
are
homomorphisms.
To
prove
uniqueness,
sup-
pose
that
we
have
a
group
structure
on
U(X
1
)
×
U(X
2
)
with
this
property.
Take
elements
(x
1
,
x
2
)
and
(x
1
0
,
x
2
0
)
of
U(X
1
)
×
U(X
2
)
and
write
(x
1
,
x
2
)
·
(x
1
0
,
x
2
0
)
=
(y
1
,
y
2
).
Since
p
1
is
a
homomorphism,
y
1
=
p
1
(y
1
,
y
2
)
=
p
1
((x
1
,
x
2
)
·
(x
1
0
,
x
2
0
))
=
p
1
(x
1
,
x
2
)
·
p
1
(x
1
0
,
x
2
0
)
=
x
1
·
x
1
0
,
and
similarly
y
2
=
x
2
·
x
2
0
.
Hence
(x
1
,
x
2
)
·
(x
1
0
,
x
2
0
)
=
(x
1
x
1
0
,
x
2
x
2
0
).
A
similar
argument
shows
that
(x
1
,
x
2
)
−1
=
(x
1
−1
,
x
2
−1
)
and
that
the
identity
element
1
of
the
group
is
(1,
1).
Now,
for
existence,
define
·,
(
)
−1
and
1
by
the
formulas
just
given;
it
can
then
be
checked
that
the
group
axioms
are
satisfied
and
that
p
1
and
p
2
are
group
homomorphisms.
This
proves
the
claim.
Write
L
for
the
set
U(X
1
)
×
U(X
2
)
equipped
with
this
group
structure.
Then
we
have
a
cone
p
1
p
2
X
1
←−
L
−→
X
2
in
Grp.
It
is
easy
to
check
that
this
is,
in
fact,
a
product
cone
in
Grp.
We
can
summarize
this
in
language
that
is
not
tied
to
group
theory.
Given
objects
X
1
and
X
2
of
Grp,
5.3
Interactions
between
functors
and
limits
A
p
D
A
F
◦D
B
139
F
B
q
Figure
5.3
Creation
of
limits.
•
for
any
product
cone
on
(U(X
1
),
U(X
2
))
in
Set,
there
is
a
unique
cone
on
(X
1
,
X
2
)
in
Grp
whose
image
under
U
is
the
cone
we
started
with;
•
this
cone
on
(X
1
,
X
2
)
is
a
product
cone.
This
suggests
the
following
definition
(Figure
5.3).
Definition
5.3.5
A
functor
F
:
A
→
B
creates
limits
(of
shape
I)
if
when-
ever
D
:
I
→
A
is
a
diagram
in
A
,
q
I
•
for
any
limit
cone
B
−→
FD(I)
on
the
diagram
F
◦
D,
there
is
a
unique
I∈I
p
I
cone
A
−→
D(I)
on
D
such
that
F(A)
=
B
and
F(p
I
)
=
q
I
for
all
I
∈
I;
p
I
I∈I
•
this
cone
A
−→
D(I)
is
a
limit
cone
on
D.
I∈I
The
forgetful
functors
from
Grp,
Ring,
.
.
.
to
Set
all
create
limits
(Exer-
cise
5.3.11).
The
word
creates
is
explained
by
the
following
result.
Lemma
5.3.6
Let
F
:
A
→
B
be
a
functor
and
I
a
small
category.
Suppose
that
B
has,
and
F
creates,
limits
of
shape
I.
Then
A
has,
and
F
preserves,
limits
of
shape
I.
Proof
Exercise
5.3.12.
Since
Set
has
all
limits,
it
follows
that
all
our
categories
of
algebras
have
all
limits,
and
that
the
forgetful
functors
preserve
them.
Remark
5.3.7
There
is
something
suspicious
about
Definition
5.3.5.
It
refers
to
equality
of
objects
of
a
category,
a
relation
that,
as
we
saw
on
page
31,
is
usually
too
strict
to
be
appropriate.
It
is
almost
always
better
to
replace
equality
by
isomorphism.
If
we
replace
equality
by
isomorphism
throughout
the
definition
of
‘creates
limits’,
we
obtain
a
more
healthy
and
inclusive
notion.
140
Limits
In
the
notation
of
Definition
5.3.5,
we
ask
that
if
F
◦
D
has
a
limit
then
there
exists
a
cone
on
D
whose
image
under
F
is
a
limit
cone,
and
that
every
such
cone
is
itself
a
limit
cone.
In
fact,
what
we
are
calling
creation
of
limits
should
really
be
called
strict
creation
of
limits,
with
‘creation
of
limits’
reserved
for
the
more
inclusive
no-
tion.
That
is
how
‘creates’
is
used
in
most
of
the
literature.
I
have
chosen
to
use
the
strict
version
here
because
it
is
slightly
simpler
to
state,
and
because
the
examples
at
hand
all
satisfy
the
stricter
condition.
Exercises
5.3.8
Taking
the
limit
is
a
process
that
receives
as
its
input
a
diagram
in
a
category
A
,
and
produces
as
its
output
a
new
object
of
A
.
Later,
we
will
see
that
this
process
is
functorial
(Proposition
6.1.4).
Here
you
are
asked
to
prove
this
in
the
case
of
binary
products.
Let
A
be
a
category
with
binary
products.
Suppose
that
we
have
chosen
for
each
pair
(X,
Y)
of
objects
a
product
cone
X
o
p
1
X,Y
X
×
Y
p
2
X,Y
/
Y.
Construct
a
functor
A
×
A
→
A
given
on
objects
by
(X,
Y)
7→
X
×
Y.
5.3.9
Let
A
be
a
category
with
binary
products.
Prove
directly
that
A
(A,
X
×
Y)
A
(A,
X)
×
A
(A,
Y)
naturally
in
A,
X,
Y
∈
A
.
(This
presupposes
that
we
have
chosen
for
each
X
and
Y
a
product
cone
on
(X,
Y).
By
Exercise
5.3.8,
the
assignment
(X,
Y)
7→
X
×
Y
is
then
functorial,
which
it
must
be
in
order
for
‘naturally’
to
make
sense.)
5.3.10
Prove
that
if
a
functor
creates
limits
then
it
also
reflects
them.
5.3.11
It
was
shown
in
Example
5.3.4
that
the
forgetful
functor
U
:
Grp
→
Set
creates
binary
products.
(a)
Using
the
formula
for
limits
in
Set
(Example
5.1.22),
prove
that,
in
fact,
U
creates
arbitrary
limits.
(b)
Satisfy
yourself
that
the
same
is
true
if
Grp
is
replaced
by
any
other
cate-
gory
of
algebras
such
as
Ring,
Ab
or
Vect
k
.
5.3.12
Prove
Lemma
5.3.6.
5.3.13
(a)
An
object
P
of
a
category
B
is
projective
if
B(P,
−)
:
B
→
Set
preserves
epics.
(This
means
that
if
f
is
epic
then
so
is
B(P,
f
).)
Let
5.3
Interactions
between
functors
and
limits
Set
o
F
⊥
/
141
B
be
an
adjunction
in
which
G
preserves
epics.
Prove
that
F(S
)
G
is
projective
for
all
sets
S
.
(b)
Find
a
non-projective
object
of
Ab.
(c)
An
object
I
of
a
category
B
is
injective
if
it
is
projective
in
B
op
,
or
equiv-
alently
if
B(−,
I)
:
B
op
→
Set
preserves
epics.
Show
that
all
objects
of
Vect
k
are
injective,
and
find
a
non-injective
object
of
Ab.
6
Adjoints,
representables
and
limits
We
have
approached
the
idea
of
universal
property
from
three
different
angles,
producing
three
different
formalisms:
adjointness,
representability,
and
limits.
In
this
final
chapter,
we
work
out
the
connections
between
them.
In
principle,
anything
that
can
be
described
in
one
of
the
three
formalisms
can
also
be
described
in
the
others.
The
situation
is
similar
to
that
of
cartesian
and
polar
coordinates:
anything
that
can
be
done
in
polar
coordinates
can
in
principle
be
done
in
cartesian
coordinates,
and
vice
versa,
but
some
things
are
more
gracefully
done
in
one
system
than
the
other.
In
comparing
the
three
approaches,
we
will
discover
many
of
the
fundamen-
tal
results
of
category
theory.
Here
are
some
highlights.
•
Limits
and
colimits
in
functor
categories
work
in
the
simplest
possible
way.
•
The
embedding
of
a
category
A
into
its
presheaf
category
[A
op
,
Set]
pre-
serves
limits
(but
not
colimits).
•
The
representables
are
the
prime
numbers
of
presheaves:
every
presheaf
can
be
expressed
canonically
as
a
colimit
of
representables.
•
A
functor
with
a
left
adjoint
preserves
limits.
Under
suitable
hypotheses,
the
converse
holds
too.
•
Categories
of
presheaves
[A
op
,
Set]
behave
very
much
like
the
category
of
sets,
the
beginning
of
an
incredible
story
that
brings
together
the
subjects
of
logic
and
geometry.
6.1
Limits
in
terms
of
representables
and
adjoints
There
is
more
than
one
way
to
present
the
definition
of
limit.
In
Chapter
5,
we
used
an
explicit
form
of
the
definition
that
is
particularly
convenient
for
examples.
But
we
will
soon
be
developing
the
theory
of
limits
and
colimits,
142
143
6.1
Limits
in
terms
of
representables
and
adjoints
and
for
that,
a
rephrased
form
of
the
definition
is
useful.
In
fact,
we
rephrase
it
in
two
different
ways:
once
in
terms
of
representability,
and
once
in
terms
of
adjoints.
We
begin
by
showing
that
cones
are
simply
natural
transformations
of
a
special
kind.
To
do
this,
we
need
some
notation.
Given
categories
I
and
A
and
an
object
A
∈
A
,
there
is
a
functor
∆A
:
I
→
A
with
constant
value
A
on
objects
and
1
A
on
maps.
This
defines,
for
each
I
and
A
,
the
diagonal
functor
∆
:
A
→
[I,
A
].
The
name
can
be
understood
by
considering
the
case
in
which
I
is
the
discrete
category
with
two
objects;
then
[I,
A
]
=
A
×
A
and
∆(A)
=
(A,
A).
Now,
given
a
diagram
D
:
I
→
A
and
an
object
A
∈
A
,
a
cone
on
D
with
vertex
A
is
simply
a
natural
transformation
∆A
I
(
8
A
.
D
Writing
Cone(A,
D)
for
the
set
of
cones
on
D
with
vertex
A,
we
therefore
have
Cone(A,
D)
=
[I,
A
](∆A,
D).
(6.1)
Thus,
Cone(A,
D)
is
functorial
in
A
(contravariantly)
and
D
(covariantly).
Here
is
our
first
rephrasing
of
the
definition
of
limit.
Proposition
6.1.1
Let
I
be
a
small
category,
A
a
category,
and
D
:
I
→
A
a
diagram.
Then
there
is
a
one-to-one
correspondence
between
limit
cones
on
D
and
representations
of
the
functor
Cone(−,
D)
:
A
op
→
Set,
with
the
representing
objects
of
Cone(−,
D)
being
the
limit
objects
(that
is,
the
vertices
of
the
limit
cones)
of
D.
Briefly
put:
a
limit
of
D
is
a
representation
of
[I,
A
](∆−,
D).
Proof
By
Corollary
4.3.2,
a
representation
of
Cone(−,
D)
consists
of
a
cone
on
D
with
a
certain
universal
property.
This
is
exactly
the
universal
property
in
the
definition
of
limit
cone.
The
proposition
formalizes
the
thought
that
cones
on
a
diagram
D
corre-
spond
one-to-one
with
maps
into
lim
D.
It
implies
that
if
D
has
a
limit
then
←I
Cone(A,
D)
A
A,
lim
D
←I
(6.2)
144
Adjoints,
representables
and
limits
3
1
/
lim
D
A
D
+
-
(b)
s
lim
α
(a)
α
1
3
/
lim
D
0
A
0
D
0
+
-
Figure
6.1
Illustration
of
Lemma
6.1.3.
naturally
in
A.
The
correspondence
is
given
from
left
to
right
by
(
f
I
)
I∈I
7→
f
¯
(in
the
notation
of
Definition
5.1.19),
and
from
right
to
left
by
(p
I
◦
g)
I∈I
→
7
g
where
p
I
:
lim
D
→
D(I)
are
the
projections.
←I
From
Proposition
6.1.1
and
Corollary
4.3.10
we
deduce:
Corollary
6.1.2
Limits
are
unique
up
to
isomorphism.
The
characterization
(6.1)
of
cones
suggests
that
we
might
consider
varying
the
diagram
D
as
well
as
the
vertex
A.
We
are
naturally
led
to
ask
questions
such
as:
given
a
map
D
→
D
0
between
diagrams,
is
there
an
induced
map
between
the
limits
of
D
and
D
0
?
The
answer
is
yes
(Figure
6.1):
D
Lemma
6.1.3
Let
I
be
a
small
category
and
I
α
$
A
<
a
natural
transfor-
0
D
mation.
Let
p
I
lim
D
−→
D(I)
←I
and
I∈I
p
0
I
lim
D
0
−→
D
0
(I)
←I
I∈I
be
limit
cones.
Then:
(a)
there
is
a
unique
map
lim
α
:
lim
D
→
lim
D
0
such
that
for
all
I
∈
I,
the
←I
←I
←I
145
6.1
Limits
in
terms
of
representables
and
adjoints
square
/
D(I)
p
I
lim
D
←I
lim
α
α
I
←I
lim
D
0
/
D
0
(I)
p
0
I
←I
commutes;
f
I
0
f
I
(b)
given
cones
A
−→
D(I)
and
A
0
−→
D
0
(I)
and
a
map
s
:
A
→
A
0
I∈I
I∈I
such
that
f
I
A
/
D(I)
α
I
s
A
0
f
I
0
/
D
0
(I)
commutes
for
all
I
∈
I,
the
square
A
f
¯
/
lim
D
←I
lim
α
s
A
0
←I
f
0
/
lim
D
0
←I
also
commutes.
Proof
α
I
p
I
Part
(a)
follows
immediately
from
the
fact
that
lim
D
−→
D
0
(I)
←I
0
a
cone
on
D
.
To
prove
(b),
note
that
for
each
I
∈
I,
we
have
p
0
I
◦
lim
α
◦
f
¯
=
α
I
◦
p
I
◦
f
¯
=
α
I
◦
f
I
=
f
I
0
◦
s
=
p
0
I
◦
f
0
◦
s.
is
I∈I
←I
So
by
Exercise
5.1.36(a),
lim
α
◦
f
¯
=
f
0
◦
s.
←I
We
can
now
give
the
second
rephrasing
of
the
definition
of
limit.
It
only
applies
when
the
category
has
all
limits
of
the
shape
concerned.
Proposition
6.1.4
Let
I
be
a
small
category
and
A
a
category
with
all
limits
of
shape
I.
Then
lim
defines
a
functor
[I,
A
]
→
A
,
and
this
functor
is
right
←I
adjoint
to
the
diagonal
functor.
Proof
Choose
for
each
D
∈
[I,
A
]
a
limit
cone
on
D,
and
call
its
vertex
lim
D.
For
each
map
α
:
D
→
D
0
in
[I,
A
],
we
have
a
canonical
map
lim
α
:
←I
←I
146
Adjoints,
representables
and
limits
lim
D
→
lim
D
0
,
defined
as
in
Lemma
6.1.3(a).
This
makes
lim
into
a
functor.
←I
←I
←I
Proposition
6.1.1
implies
that
[I,
A
](∆A,
D)
=
Cone(A,
D)
A
A,
lim
D
←I
naturally
in
A,
and
taking
s
=
1
A
in
Lemma
6.1.3(b)
tells
us
that
the
isomor-
phism
is
also
natural
in
D.
To
define
the
functor
lim
,
we
had
to
choose
for
each
D
a
limit
cone
on
D.
←I
This
is
a
non-canonical
choice.
Nevertheless,
different
choices
only
affect
the
functor
lim
up
to
natural
isomorphism,
by
uniqueness
of
adjoints.
←I
Exercises
6.1.5
Interpret
all
the
theory
of
this
section
in
the
special
case
where
I
is
the
discrete
category
with
two
objects.
6.1.6
What
is
the
content
of
Proposition
6.1.4
when
I
is
a
group
and
A
=
Set?
What
about
the
dual
of
Proposition
6.1.4?
6.2
Limits
and
colimits
of
presheaves
What
do
limits
and
colimits
look
like
in
functor
categories
[A
,
B]?
In
par-
ticular,
what
do
they
look
like
in
presheaf
categories
[A
op
,
Set]?
More
par-
ticularly
still,
what
about
limits
and
colimits
of
representables?
Are
they,
too,
representable?
We
will
answer
all
these
questions.
In
order
to
do
so,
we
first
prove
that
representables
preserve
limits.
Representables
preserve
limits
Let
us
begin
by
recalling
that,
by
definition
of
product,
a
map
A
→
X
×
Y
amounts
to
a
pair
of
maps
(A
→
X,
A
→
Y).
Here
A,
X
and
Y
are
objects
of
a
category
A
with
binary
products.
There
is,
therefore,
a
bijection
A
(A,
X
×
Y)
A
(A,
X)
×
A
(A,
Y)
(6.3)
natural
in
A,
X,
Y
∈
A
.
Is
this
a
special
feature
of
products,
or
does
some
analogous
statement
hold
for
every
kind
of
limit?
Let
us
try
equalizers.
Suppose
that
A
has
equalizers,
and
write
Eq
X
6.2
Limits
and
colimits
of
presheaves
147
s
/
/
Y
for
the
equalizer
of
maps
s
and
t.
By
definition
of
t
equalizer,
maps
s
A
→
Eq
X
t
/
/
Y
(6.4)
correspond
one-to-one
with
maps
f
:
A
→
X
such
that
s
◦
f
=
t
◦
f
.
Now
recall
that
s
induces
a
map
s
∗
=
A
(A,
s)
:
A
(A,
X)
→
A
(A,
Y),
and
similarly
for
t.
In
this
notation,
what
we
have
just
said
is
that
maps
(6.4)
correspond
one-to-one
with
elements
f
∈
A
(A,
X)
such
that
A
(A,
s)
(
f
)
=
A
(A,
t)
(
f
).
By
the
explicit
formula
for
equalizers
in
Set
(Example
5.1.12),
such
an
f
is
exactly
an
element
of
the
equalizer
of
A
(A,
s)
and
A
(A,
t).
So,
we
have
a
canonical
bijection
s
/
/
Y
Eq
A
(A,
X)
A
(A,s)
/
/
A
(A,
Y)
.
(6.5)
A
A,
Eq
X
A
(A,t)
t
This
looks
something
like
our
isomorphism
(6.3)
for
products.
The
isomorphisms
(6.3)
and
(6.5)
suggest
that,
more
generally,
we
might
have
A
A,
lim
D
lim
A
(A,
D)
←I
←I
(6.6)
naturally
in
A
∈
A
and
D
∈
[I,
A
],
whenever
A
is
a
category
with
limits
of
shape
I.
Here
A
(A,
D)
is
the
functor
A
(A,
D)
:
I
I
→
Set
7
→
A
(A,
D(I)).
This
functor
could
also
be
written
as
A
(A,
D(−)),
and
is
the
composite
I
D
/
A
A
(A,−)
/
Set.
The
conjectured
isomorphism
(6.6)
states,
essentially,
that
representables
pre-
serve
limits.
We
now
set
about
proving
this.
Lemma
6.2.1
Let
I
be
a
small
category,
A
a
locally
small
category,
D
:
I
→
A
a
diagram,
and
A
∈
A
.
Then
Cone(A,
D)
lim
A
(A,
D)
←I
naturally
in
A
and
D.
148
Adjoints,
representables
and
limits
Proof
Like
all
functors
from
a
small
category
into
Set,
the
functor
A
(A,
D)
does
have
a
limit,
given
by
the
explicit
formula
(5.16).
According
to
this
for-
mula,
lim
A
(A,
D)
is
the
set
of
all
families
(
f
I
)
I∈I
such
that
f
I
∈
A
(A,
D(I))
←I
for
all
I
∈
I
and
(A
(A,
Du))(
f
I
)
=
f
J
(6.7)
u
for
all
I
−→
J
in
I.
But
equation
(6.7)
just
says
that
(Du)
◦
f
I
=
f
J
,
so
an
element
of
lim
A
(A,
D)
is
nothing
but
a
cone
on
D
with
vertex
A.
←I
Proposition
6.2.2
(Representables
preserve
limits)
Let
A
be
a
locally
small
category
and
A
∈
A
.
Then
A
(A,
−)
:
A
→
Set
preserves
limits.
Proof
Let
I
be
a
small
category
and
let
D
:
I
→
A
be
a
diagram
that
has
a
limit.
Then
A
A,
lim
D
Cone(A,
D)
lim
A
(A,
D)
←I
←I
naturally
in
A.
Here
the
first
isomorphism
is
Proposition
6.1.1
(or
more
partic-
ularly,
the
isomorphism
(6.2)
that
follows
it),
and
the
second
is
Lemma
6.2.1.
Remark
6.2.3
Proposition
6.2.2
tells
us
that
A
A,
lim
D
lim
A
(A,
D).
←I
←I
(6.8)
To
dualize
Proposition
6.2.2,
we
replace
A
by
A
op
.
Thus,
A
(−,
A)
:
A
op
→
Set
preserves
limits.
A
limit
in
A
op
is
a
colimit
in
A
,
so
A
(−,
A)
transforms
colimits
in
A
into
limits
in
Set:
A
lim
D,
A
lim
op
A
(D,
A).
(6.9)
→I
←I
The
right-hand
side
is
a
limit,
not
a
colimit!
So
even
though
(6.8)
and
(6.9)
are
dual
statements,
there
are,
in
total,
more
limits
than
colimits
involved.
Some-
how,
limits
have
the
upper
hand.
For
example,
let
X,
Y
and
A
be
objects
of
a
category
A
,
and
suppose
that
the
sum
X
+
Y
exists.
By
definition
of
sum,
a
map
X
+
Y
→
A
amounts
to
a
pair
of
maps
(X
→
A,
Y
→
A).
In
other
words,
there
is
a
canonical
isomorphism
A
(X
+
Y,
A)
A
(X,
A)
×
A
(Y,
A).
This
is
the
isomorphism
(6.9)
in
the
case
where
I
is
the
discrete
category
with
two
objects.
6.2
Limits
and
colimits
of
presheaves
149
Limits
in
functor
categories
Earlier,
we
learned
that
it
is
sometimes
useful
to
view
functors
as
objects
in
their
own
right,
rather
than
as
maps
of
categories.
For
instance,
when
G
is
a
group,
functors
G
→
Set
are
G-sets
(Example
1.2.8),
which
one
would
usually
regard
as
‘things’
rather
than
‘maps’.
This
point
of
view
leads
to
the
concept
of
functor
category.
We
now
begin
an
analysis
of
limits
and
colimits
in
functor
categories
[A,
S
].
Here
A
is
small
and
S
is
locally
small;
these
conditions
together
guarantee
that
[A,
S
]
is
locally
small.
The
most
important
cases
for
us
will
be
S
=
Set
and
S
=
Set
op
.
For
that
reason,
we
will
assume
whenever
necessary
that
S
has
all
limits
and
colimits.
We
show
that
limits
and
colimits
in
[A,
S
]
work
in
the
simplest
way
imag-
inable.
For
instance,
if
S
has
binary
products
then
so
does
[A,
S
],
and
the
product
of
two
functors
X,
Y
:
A
→
S
is
the
functor
X
×
Y
:
A
→
S
given
by
(X
×
Y)(A)
=
X(A)
×
Y(A)
for
all
A
∈
A.
Notation
6.2.4
Let
A
and
S
be
categories.
For
each
A
∈
A,
there
is
a
functor
ev
A
:
[A,
S
]
X
→
7→
S
X(A),
called
evaluation
at
A.
We
will
be
working
with
diagrams
in
[A,
S
],
and
given
such
a
diagram
D
:
I
→
[A,
S
],
we
have
for
each
A
∈
A
a
functor
ev
A
◦D
:
I
I
→
7
→
S
D(I)(A).
We
write
ev
A
◦D
as
D(−)(A).
Theorem
6.2.5
(Limits
in
functor
categories)
Let
A
and
I
be
small
cate-
gories
and
S
a
locally
small
category.
Let
D
:
I
→
[A,
S
]
be
a
diagram,
and
suppose
that
for
each
A
∈
A,
the
diagram
D(−)(A)
:
I
→
S
has
a
limit.
Then
there
is
a
cone
on
D
whose
image
under
ev
A
is
a
limit
cone
on
D(−)(A)
for
each
A
∈
A.
Moreover,
any
such
cone
on
D
is
a
limit
cone.
Theorem
6.2.5
is
often
expressed
as
a
slogan:
Limits
in
a
functor
category
are
computed
pointwise.
The
‘points’
in
the
word
‘pointwise’
are
the
objects
of
A.
The
slogan
means,
for
example,
that
given
two
functors
X,
Y
∈
[A,
S
],
their
product
can
be
computed
150
Adjoints,
representables
and
limits
by
first
taking
the
product
X(A)×Y(A)
in
S
for
each
‘point’
A,
then
assembling
them
to
form
a
functor
X
×
Y.
Of
course,
Theorem
6.2.5
has
a
dual,
stating
that
colimits
in
a
functor
cate-
gory
are
also
computed
pointwise.
Take
for
each
A
∈
A
a
limit
cone
p
I,A
L(A)
−→
D(I)(A)
Proof
of
Theorem
6.2.5
I∈I
(6.10)
on
the
diagram
D(−)(A)
:
I
→
S
.
We
prove
two
statements:
(a)
there
is
exactly
one
way
of
extending
L
to
a
functor
on
A
with
the
property
p
I
that
L
−→
D(I)
is
a
cone
on
D;
p
I
I∈I
(b)
this
cone
L
−→
D(I)
is
a
limit
cone.
I∈I
The
theorem
will
follow
immediately.
For
(a),
take
a
map
f
:
A
→
A
0
in
A.
Lemma
6.1.3(a)
applied
to
the
natural
transformation
D(−)(A)
I
D(−)(
f
)
(
6
S
D(−)(A
0
)
implies
that
there
is
a
unique
map
L(
f
)
:
L(A)
→
L(A
0
)
such
that
for
all
I
∈
I,
the
square
L(A)
/
D(I)(A)
p
I,A
L(
f
)
L(A
0
)
D(I)(
f
)
/
D(I)(A
0
)
p
I,A
0
(6.11)
commutes.
(This
is
our
definition
of
L(
f
).)
We
have
now
defined
L
on
objects
and
maps
of
A.
It
is
easy
to
check
that
L
preserves
composition
and
identities,
and
is
therefore
a
functor
L
:
A
→
S
.
Moreover,
the
commutativity
of
dia-
p
I,A
gram
(6.11)
says
exactly
that
for
each
I
∈
I,
the
family
L(A)
−→
D(I)(A)
A∈A
is
a
natural
transformation
L
A
p
I
(
6
S
.
D(I)
p
I
So
we
have
a
family
L
−→
D(I)
of
maps
in
[A,
S
],
and
from
the
fact
I∈I
that
(6.10)
is
a
cone
on
D(−)(A)
for
each
A
∈
A,
it
follows
immediately
that
p
I
is
a
cone
on
D.
L
−→
D(I)
I∈I
6.2
Limits
and
colimits
of
presheaves
151
q
I
For
(b),
let
X
∈
[A,
S
]
and
let
X
−→
D(I)
be
a
cone
on
D
in
[A,
S
].
I∈I
For
each
A
∈
A,
we
have
a
cone
q
I,A
X(A)
−→
D(I)(A)
I∈I
on
D(−)(A)
in
S
,
so
there
is
a
unique
map
q̄
A
:
X(A)
→
L(A)
such
that
p
I,A
◦
q̄
A
=
q
I,A
for
all
I
∈
I.
It
only
remains
to
prove
that
q̄
A
is
natural
in
A,
and
that
follows
from
Lemma
6.1.3(b).
Theorem
6.2.5
has
many
important
consequences.
We
begin
by
recording
a
cruder
form
of
the
theorem
(and
its
dual),
which
we
will
use
repeatedly.
Corollary
6.2.6
Let
I
and
A
be
small
categories,
and
S
a
locally
small
cate-
gory.
If
S
has
all
limits
(respectively,
colimits)
of
shape
I
then
so
does
[A,
S
],
and
for
each
A
∈
A,
the
evaluation
functor
ev
A
:
[A,
S
]
→
S
preserves
them.
Warning
6.2.7
If
S
does
not
have
all
limits
of
shape
I
then
[A,
S
]
may
con-
tain
limits
of
shape
I
that
are
not
computed
pointwise,
that
is,
are
not
preserved
by
all
the
evaluation
functors.
Examples
can
be
constructed,
as
in
Section
3.3
of
Kelly
(1982).
Theorem
6.2.5
will
also
help
us
to
prove
that
limits
commute
with
limits,
in
the
following
sense.
Take
categories
I,
J
and
S
.
There
are
isomorphisms
of
categories
[I,
[J,
S
]]
[I
×
J,
S
]
[J,
[I,
S
]].
(See
Remark
4.1.23(c)
and
Exercise
1.2.25.)
Under
these
isomorphisms,
a
functor
D
:
I
×
J
→
S
corresponds
to
the
functors
D
•
:
I
I
→
7
→
[J,
S
]
D(I,
−)
and
D
•
:
J
J
→
7
→
[I,
S
]
D(−,
J).
Supposing
that
S
has
all
limits,
so
do
the
various
functor
categories,
by
Corol-
lary
6.2.6.
In
particular,
there
is
an
object
lim
D
•
of
[J,
S
].
This
is
itself
a
dia-
←I
gram
in
S
,
so
we
obtain
in
turn
an
object
lim
lim
D
•
of
S
.
Alternatively,
we
←J
←I
can
take
limits
in
the
other
order,
producing
an
object
lim
lim
D
•
of
S
.
And
←I
←J
there
is
a
third
possibility:
taking
the
limit
of
D
itself,
we
obtain
another
object
lim
D
of
S
.
The
next
result
states
that
these
three
objects
are
the
same.
That
←I×J
is,
it
makes
no
difference
what
order
we
take
limits
in.
Proposition
6.2.8
(Limits
commute
with
limits)
Let
I
and
J
be
small
cate-
gories.
Let
S
be
a
locally
small
category
with
limits
of
shape
I
and
of
shape
152
Adjoints,
representables
and
limits
J.
Then
for
all
D
:
I
×
J
→
S
,
we
have
lim
lim
D
•
lim
D
lim
lim
D
•
,
←J
←I
←I×J
←I
←J
and
all
these
limits
exist.
In
particular,
S
has
limits
of
shape
I
×
J.
This
is
sometimes
half-jokingly
called
Fubini’s
theorem,
as
it
is
something
like
changing
the
order
of
integration
in
a
double
integral.
The
analogy
is
more
appealing
with
colimits,
since,
like
integrals,
colimits
can
be
thought
of
as
a
context-sensitive
version
of
sums.
Proof
By
symmetry,
it
is
enough
to
prove
the
first
isomorphism.
Since
S
has
limits
of
shape
I,
so
does
[J,
S
]
(by
Corollary
6.2.6).
So
lim
D
•
exists;
it
←I
is
an
object
of
[J,
S
].
Since
S
has
limits
of
shape
J,
lim
lim
D
•
exists;
it
is
an
←J
←I
object
of
S
.
Then
for
S
∈
S
,
S
S
,
lim
lim
D
•
[J,
S
]
∆S
,
lim
D
•
←J
←I
←I
[I,
[J,
S
]](∆(∆S
),
D
•
)
[I
×
J,
S
](∆S
,
D)
naturally
in
S
.
The
first
two
steps
each
follow
from
Proposition
6.1.1.
The
third
uses
the
isomorphism
[I,
[J,
S
]]
[I×J,
S
],
under
which
∆(∆S
)
corresponds
to
∆S
and
D
•
corresponds
to
D.
Hence
lim
lim
D
•
is
a
representing
object
for
the
functor
[I
×
J,
S
](∆−,
D).
←J
←I
By
Proposition
6.1.1
again,
this
says
that
lim
D
exists
and
is
isomorphic
to
←I×J
lim
lim
D
•
.
←J
←I
•
,
Proposition
6.2.8
says
that
bi-
Example
6.2.9
When
I
=
J
=
•
nary
products
commute
with
binary
products:
if
S
has
binary
products
and
Q
S
11
,
S
12
,
S
21
,
S
22
∈
S
then
the
4-fold
product
i,
j∈{1,2}
S
i
j
exists
and
satisfies
Y
(S
11
×
S
21
)
×
(S
12
×
S
22
)
S
i
j
(S
11
×
S
12
)
×
(S
21
×
S
22
).
i,
j∈{1,2}
More
generally,
it
makes
no
difference
what
order
we
write
products
in
or
where
we
put
the
brackets:
there
are
canonical
isomorphisms
S
×
T
T
×
S
,
(S
×
T
)
×
U
S
×
(T
×
U)
in
any
category
with
binary
products.
If
there
is
also
a
terminal
object
1,
there
are
further
canonical
isomorphisms
S
×
1
S
1
×
S
.
153
6.2
Limits
and
colimits
of
presheaves
Warning
6.2.10
The
dual
of
Proposition
6.2.8
states
that
colimits
commute
with
colimits.
For
instance,
(S
11
+
S
21
)
+
(S
12
+
S
22
)
(S
11
+
S
12
)
+
(S
21
+
S
22
)
in
any
category
S
with
binary
sums.
But
limits
do
not
in
general
commute
with
colimits.
For
instance,
in
general,
(S
11
+
S
21
)
×
(S
12
+
S
22
)
6
(S
11
×
S
12
)
+
(S
21
×
S
22
).
A
counterexample
is
given
by
taking
S
=
Set
and
each
S
i
j
to
be
a
one-element
set.
Then
the
left-hand
side
has
(1
+
1)
×
(1
+
1)
=
4
elements,
whereas
the
right-hand
side
has
(1
×
1)
+
(1
×
1)
=
2
elements.
Here
are
two
further
consequences
of
Theorem
6.2.5.
Corollary
6.2.11
Let
A
be
a
small
category.
Then
[A
op
,
Set]
has
all
limits
and
colimits,
and
for
each
A
∈
A,
the
evaluation
functor
ev
A
:
[A
op
,
Set]
→
Set
preserves
them.
Proof
Since
Set
has
all
limits
and
colimits,
this
is
immediate
from
Corol-
lary
6.2.6.
Corollary
6.2.12
The
Yoneda
embedding
H
•
:
A
→
[A
op
,
Set]
preserves
lim-
its,
for
any
small
category
A.
p
I
Proof
Let
D
:
I
→
A
be
a
diagram
in
A,
and
let
lim
D
−→
D(I)
be
a
←I
I∈I
limit
cone.
For
each
A
∈
A,
the
composite
functor
H
•
ev
A
A
−→
[A
op
,
Set]
−→
Set
is
H
A
,
which
preserves
limits
(Proposition
6.2.2).
So
for
each
A
∈
A,
ev
H
(p
)
A
•
I
/
ev
A
H
•
(D(I))
ev
A
H
•
lim
D
←I
I∈I
is
a
limit
cone.
But
then,
by
the
‘moreover’
part
of
Theorem
6.2.5
applied
to
the
diagram
H
•
◦
D
in
[A
op
,
Set],
the
cone
H
(p
)
•
I
H
•
lim
D
−→
H
•
(D(I))
←I
I∈I
is
also
a
limit,
as
required.
Example
6.2.13
Let
A
be
a
category
with
binary
products.
Corollary
6.2.12
implies
that
for
all
X,
Y
∈
A,
H
X×Y
H
X
×
H
Y
(6.12)
154
Adjoints,
representables
and
limits
in
[A
op
,
Set].
When
evaluated
at
a
particular
object
A,
this
says
that
A(A,
X
×
Y)
A(A,
X)
×
A(A,
Y)
(using
the
fact
that
products
are
computed
pointwise).
This
is
the
isomor-
phism
(6.3)
that
we
met
at
the
beginning
of
this
section.
Suppose
that
we
view
A
as
a
subcategory
of
[A
op
,
Set],
identifying
A
∈
A
with
the
representable
H
A
∈
[A
op
,
Set]
as
in
Figure
4.1.
Then
the
isomor-
phism
(6.12)
means
that
given
two
objects
of
A
whose
product
we
want
to
form,
it
makes
no
difference
whether
we
think
of
the
product
as
taking
place
in
A
or
[A
op
,
Set].
Similarly,
if
A
has
all
limits,
taking
limits
does
not
help
us
to
escape
from
A
into
the
rest
of
[A
op
,
Set]:
any
limit
of
representable
presheaves
is
again
representable.
Warning
6.2.14
The
Yoneda
embedding
does
not
preserve
colimits.
For
ex-
ample,
if
A
has
an
initial
object
0
then
H
0
is
not
initial,
since
H
0
(0)
=
A(0,
0)
is
a
one-element
set,
whereas
the
initial
object
of
[A
op
,
Set]
is
the
presheaf
with
constant
value
∅.
We
investigate
colimits
of
representables
next.
Every
presheaf
is
a
colimit
of
representables
We
now
know
that
the
Yoneda
embedding
preserves
limits
but
not
colimits.
In
fact,
the
situation
for
colimits
is
at
the
opposite
extreme
from
the
situation
for
limits:
by
taking
colimits
of
representable
presheaves,
we
can
obtain
any
presheaf
we
like!
This
is
the
last
main
result
of
this
section.
Every
positive
integer
can
be
expressed
as
a
product
of
primes
in
an
essen-
tially
unique
way.
Somewhat
similarly,
every
presheaf
can
be
expressed
as
a
colimit
of
representables
in
a
canonical
(though
not
unique)
way.
The
repre-
sentables
are
the
building
blocks
of
presheaves.
For
a
different
analogy,
recall
that
any
complex
function
holomorphic
in
a
neighbourhood
of
0
has
a
power
series
expansion,
such
as
e
z
=
1
+
z
+
z
2
z
3
+
+
···
.
2!
3!
In
this
sense,
the
power
functions
z
7→
z
n
are
the
building
blocks
of
holo-
morphic
functions.
We
could
even
take
the
analogy
further:
(
)
n
is
like
a
rep-
resentable
Hom(n,
−),
and
in
the
categorical
context,
quotients
and
sums
are
types
of
colimit.
Before
we
state
and
prove
the
theorem,
let
us
look
at
an
easy
special
case.
Example
6.2.15
Let
A
be
the
discrete
category
with
two
objects,
K
and
L.
A
6.2
Limits
and
colimits
of
presheaves
155
presheaf
X
on
A
is
just
a
pair
(X(K),
X(L))
of
sets,
and
[A
op
,
Set]
Set
×
Set.
There
are
two
representables,
H
K
and
H
L
,
given
by
1
if
A
=
B,
H
A
(B)
=
A(B,
A)
∅
if
A
,
B
(A,
B
∈
{K,
L}).
Identifying
[A
op
,
Set]
with
Set
×
Set,
we
have
H
K
(1,
∅)
and
H
L
(∅,
1).
Every
object
of
Set
×
Set
is
a
sum
of
copies
of
(1,
∅)
and
(∅,
1).
Suppose,
for
instance,
that
X(K)
has
three
elements
and
X(L)
has
two
elements.
Then
(X(K),
X(L))
(1,
∅)
+
(1,
∅)
+
(1,
∅)
+
(∅,
1)
+
(∅,
1)
in
Set
×
Set.
Equivalently,
X
H
K
+
H
K
+
H
K
+
H
L
+
H
L
in
[A
op
,
Set],
exhibiting
X
as
a
sum
of
representables.
In
this
example,
X
is
expressed
as
a
sum
of
five
representables,
that
is,
a
sum
indexed
by
the
set
X(K)
+
X(L)
of
‘elements’
of
X.
A
sum
is
a
colimit
over
a
discrete
category.
In
the
general
case,
a
presheaf
X
on
a
category
A
is
expressed
as
a
colimit
over
a
category
whose
objects
can
be
thought
of
as
the
‘elements’
of
X.
This
is
made
precise
by
the
following
definition.
Definition
6.2.16
Let
A
be
a
category
and
X
a
presheaf
on
A.
The
category
of
elements
E(X)
of
X
is
the
category
in
which:
•
objects
are
pairs
(A,
x)
with
A
∈
A
and
x
∈
X(A);
•
maps
(A
0
,
x
0
)
→
(A,
x)
are
maps
f
:
A
0
→
A
in
A
such
that
(X
f
)(x)
=
x
0
.
There
is
a
projection
functor
P
:
E(X)
→
A
defined
by
P(A,
x)
=
A
and
P(
f
)
=
f
.
The
following
‘density
theorem’
states
that
every
presheaf
is
a
colimit
of
representables
in
a
canonical
way.
It
is
secretly
dual
to
the
Yoneda
lemma.
This
becomes
apparent
if
one
expresses
both
in
suitably
lofty
categorical
language
(that
of
ends,
or
that
of
bimodules);
but
that
is
beyond
the
scope
of
this
book.
Theorem
6.2.17
(Density)
Let
A
be
a
small
category
and
X
a
presheaf
on
A.
Then
X
is
the
colimit
of
the
diagram
P
H
•
E(X)
−→
A
−→
[A
op
,
Set]
in
[A
op
,
Set];
that
is,
X
lim
(H
•
◦
P).
→E(X)
156
Adjoints,
representables
and
limits
Proof
First
note
that
since
A
is
small,
so
too
is
E(X).
Hence
H
•
◦
P
really
is
a
diagram
in
our
customary
sense
(Definition
5.1.18).
Now
let
Y
∈
[A
op
,
Set].
A
cocone
on
H
•
◦
P
with
vertex
Y
is
a
family
α
A,x
H
A
−→
Y
A∈A,x∈X(A)
f
of
natural
transformations
with
the
property
that
for
all
maps
A
0
−→
A
in
A
and
all
x
∈
X(A),
the
diagram
H
A
0
α
A
0
,(X
f
)(x)
(
6
Y
H
f
H
A
α
A,x
commutes.
Equivalently
(by
the
Yoneda
lemma),
a
cocone
on
H
•
◦
P
with
vertex
Y
is
a
family
(y
A,x
)
A∈A,x∈X(A)
,
f
with
y
A,x
∈
Y(A),
such
that
for
all
maps
A
0
−→
A
in
A
and
all
x
∈
X(A),
(Y
f
)(y
A,x
)
=
y
A
0
,(X
f
)(x)
.
To
see
this,
note
that
if
α
A,x
∈
[A
op
,
Set](H
A
,
Y)
corresponds
to
y
A,x
∈
Y(A),
then
α
A,x
◦
H
f
∈
[A
op
,
Set](H
A
0
,
Y)
corresponds
to
(Y
f
)(y
A,x
)
∈
Y(A
0
).
Equivalently
(writing
y
A,x
as
ᾱ
A
(x)),
it
is
a
family
ᾱ
A
X(A)
−→
Y(A)
A∈A
f
of
functions
with
the
property
that
for
all
maps
A
0
−→
A
in
A
and
all
x
∈
X(A),
(Y
f
)
ᾱ
A
(x)
=
ᾱ
A
0
(X
f
)(x)
.
But
this
is
simply
a
natural
transformation
ᾱ
:
X
→
Y.
So
we
have,
for
each
Y
∈
[A
op
,
Set],
a
canonical
bijection
[E(X),
[A
op
,
Set]](H
•
◦
P,
∆Y)
[A
op
,
Set](X,
Y).
Hence
X
is
the
colimit
of
H
•
◦
P.
Example
6.2.18
In
Example
6.2.15,
we
expressed
a
particular
presheaf
X
as
a
sum
of
representables.
Let
us
check
that
the
way
we
did
this
is
a
special
case
of
the
general
construction
in
the
density
theorem.
Since
A
is
discrete,
the
category
of
elements
E(X)
is
also
discrete;
it
is
the
set
X(K)+X(L)
with
five
elements.
The
projection
P
:
E(X)
→
A
sends
three
of
the
6.2
Limits
and
colimits
of
presheaves
157
elements
to
K
and
the
other
two
to
L,
so
the
diagram
H
•
◦P
:
E(X)
→
[A
op
,
Set]
sends
three
of
the
elements
to
H
K
and
two
to
H
L
.
The
colimit
of
H
•
◦
P
is
the
sum
of
these
five
representables,
which
is
X,
just
as
in
Example
6.2.15.
Remarks
6.2.19
(a)
The
term
‘category
of
elements’
is
compatible
with
the
generalized
element
terminology
introduced
in
Definition
4.1.25.
A
gen-
eralized
element
of
an
object
X
is
just
a
map
into
X,
say
Z
→
X;
but,
as
explained
after
that
definition,
we
often
focus
on
certain
special
shapes
Z.
Now
suppose
that
we
are
working
in
a
presheaf
category
[A
op
,
Set].
Among
all
presheaves,
the
representables
have
a
special
status,
so
we
might
be
especially
interested
in
generalized
elements
of
representable
shape.
The
Yoneda
lemma
implies
that
for
a
presheaf
X,
the
generalized
elements
of
X
of
representable
shape
correspond
to
pairs
(A,
x)
with
A
∈
A
and
x
∈
X(A).
In
other
words,
they
are
the
objects
of
the
category
of
ele-
ments.
(b)
In
topology,
a
subspace
A
of
a
space
B
is
called
dense
if
every
point
in
B
can
be
obtained
as
a
limit
of
points
in
A.
This
provides
some
explanation
for
the
name
of
Theorem
6.2.17:
the
category
A
is
‘dense’
in
[A
op
,
Set]
because
every
object
of
[A
op
,
Set]
can
be
obtained
as
a
colimit
of
objects
of
A.
Exercises
6.2.20
Fix
a
small
category
A.
(a)
Let
S
be
a
locally
small
category
with
pullbacks.
Show
that
a
natural
transformation
X
A
α
'
8
S
Y
is
monic
(as
a
map
in
[A,
S
])
if
and
only
if
α
A
is
monic
for
all
A
∈
A.
(Hint:
use
Lemma
5.1.32.)
(b)
Describe
explicitly
the
monics
and
epics
in
[A
op
,
Set].
(c)
Can
you
do
part
(b)
without
relying
on
the
fact
that
limits
and
colimits
of
presheaves
are
computed
pointwise?
6.2.21
(a)
Prove
that
representables
have
the
following
connectedness
prop-
erty:
given
a
locally
small
category
A
and
A
∈
A
,
if
X,
Y
∈
[A
op
,
Set]
with
H
A
X
+
Y,
then
either
X
or
Y
is
the
constant
functor
∅.
(b)
Deduce
that
the
sum
of
two
representables
is
never
representable.
158
Adjoints,
representables
and
limits
6.2.22
Show
how
a
category
of
elements
can
be
described
as
a
comma
cate-
gory.
6.2.23
Let
X
be
a
presheaf
on
a
locally
small
category.
Show
that
X
is
repre-
sentable
if
and
only
if
its
category
of
elements
has
a
terminal
object.
(Since
a
terminal
object
is
a
limit
of
the
empty
diagram,
this
implies
that
the
concept
of
representability
can
be
derived
from
the
concept
of
limit.
Since
a
terminal
object
of
a
category
E
is
also
a
right
adjoint
to
the
unique
functor
E
→
1,
the
concept
of
representability
can
also
be
derived
from
the
concept
of
adjoint.)
6.2.24
Prove
that
every
slice
of
a
presheaf
category
is
again
a
presheaf
cat-
egory.
That
is,
given
a
small
category
A
and
a
presheaf
X
on
A,
prove
that
[A
op
,
Set]/X
is
equivalent
to
[B
op
,
Set]
for
some
small
category
B.
6.2.25
Let
F
:
A
→
B
be
a
functor
between
small
categories.
For
each
object
B
∈
B,
there
is
a
comma
category
(F
⇒
B)
(defined
dually
to
the
comma
category
in
Example
2.3.4),
and
there
is
a
projection
functor
P
B
:
(F⇒B)
→
A.
(a)
Let
X
:
A
→
S
be
a
functor
from
A
to
a
category
S
with
small
colimits.
For
each
B
∈
B,
let
(Lan
F
X)(B)
be
the
colimit
of
the
diagram
P
B
X
(F
⇒
B)
−→
A
−→
S
.
Show
that
this
defines
a
functor
Lan
F
X
:
B
→
S
,
and
that
for
functors
Y
:
B
→
S
,
there
is
a
canonical
bijection
between
natural
transformations
Lan
F
X
→
Y
and
natural
transformations
X
→
Y
◦
F.
(b)
Deduce
that
for
any
category
S
with
small
colimits,
the
functor
−
◦
F
:
[B,
S
]
→
[A,
S
]
has
a
left
adjoint.
(This
left
adjoint,
Lan
F
,
is
called
left
Kan
extension
along
F.)
(c)
Part
(b)
and
its
dual
imply
that
when
S
has
small
limits
and
colimits,
the
functor
−
◦
F
has
both
left
and
right
adjoints.
Revisit
Exercise
2.1.16
with
this
in
mind,
taking
F
to
be
either
the
unique
functor
1
→
G
or
the
unique
functor
G
→
1.
6.3
Interactions
between
adjoint
functors
and
limits
We
saw
in
Proposition
4.1.11
that
any
set-valued
functor
with
a
left
adjoint
is
representable,
and
in
Proposition
6.2.2
that
any
representable
preserves
limits.
6.3
Interactions
between
adjoint
functors
and
limits
159
Hence,
any
set-valued
functor
with
a
left
adjoint
preserves
limits.
In
fact,
this
conclusion
holds
not
only
for
set-valued
functors,
but
in
complete
generality.
Let
A
o
Theorem
6.3.1
F
⊥
G
/
B
be
an
adjunction.
Then
F
preserves
colimits
and
G
preserves
limits.
Proof
By
duality,
it
is
enough
to
prove
that
G
preserves
limits.
Let
D
:
I
→
B
be
a
diagram
for
which
a
limit
exists.
Then
A
A,
G
lim
D
B
F(A),
lim
D
(6.13)
←I
←I
lim
B(F(A),
D)
(6.14)
lim
A
(A,
G
◦
D)
(6.15)
Cone(A,
G
◦
D)
(6.16)
←I
←I
naturally
in
A
∈
A
.
Here,
the
isomorphism
(6.13)
is
by
adjointness,
(6.14)
is
because
representables
preserve
(6.15)
is
by
adjointness
again,
and
limits,
(6.16)
is
by
Lemma
6.2.1.
So
G
lim
D
represents
Cone(−,
G
◦
D);
that
is,
it
is
←I
a
limit
of
G
◦
D.
Example
6.3.2
Forgetful
functors
from
categories
of
algebras
to
Set
have
left
adjoints,
but
hardly
ever
right
adjoints.
Correspondingly,
they
preserve
all
limits,
but
rarely
all
colimits.
Example
6.3.3
Every
set
B
gives
rise
to
an
adjunction
(−
×
B)
a
(−)
B
of
functors
from
Set
to
Set
(Example
2.1.6).
So
−
×
B
preserves
colimits
and
(−)
B
preserves
limits.
In
particular,
−
×
B
preserves
finite
sums
and
(−)
B
preserves
finite
products,
giving
isomorphisms
0
×
B
0,
B
1
1,
(A
1
+
A
2
)
×
B
(A
1
×
B)
+
(A
2
×
B),
B
(A
1
×
A
2
)
A
1
B
×
A
2
B
.
(6.17)
(6.18)
These
are
the
analogues
of
standard
rules
of
arithmetic.
(See
also
Example
6.2.9
and
the
‘Digression
on
arithmetic’
on
page
69.)
Indeed,
if
we
know
(6.17)
and
(6.18)
for
just
finite
sets
then
by
taking
cardinality
on
both
sides,
we
ob-
tain
exactly
these
standard
rules.
The
natural
numbers
are,
after
all,
just
the
isomorphism
classes
of
finite
sets.
Example
6.3.4
adjunction
A
o
Given
a
category
A
with
all
limits
of
shape
I,
we
have
the
/
[I,
A
]
(Proposition
6.1.4).
Hence
lim
preserves
limits,
or
∆
⊥
lim
←I
←I
160
Adjoints,
representables
and
limits
equivalently,
limits
of
shape
I
commute
with
(all)
limits.
This
gives
another
proof
that
limits
commute
with
limits
(Proposition
6.2.8),
at
least
in
the
case
where
the
category
has
all
limits
of
one
of
the
shapes
concerned.
Example
6.3.5
Theorem
6.3.1
is
often
used
to
prove
that
a
functor
does
not
have
an
adjoint.
For
instance,
it
was
claimed
in
Example
2.1.3(e)
that
the
for-
getful
functor
U
:
Field
→
Set
does
not
have
a
left
adjoint.
We
can
now
prove
this.
If
U
had
a
left
adjoint
F
:
Set
→
Field,
then
F
would
preserve
colim-
its,
and
in
particular,
initial
objects.
Hence
F(∅)
would
be
an
initial
object
of
Field.
But
Field
has
no
initial
object,
since
there
are
no
maps
between
fields
of
different
characteristic.
Further
examples
of
nonexistence
of
adjoints
can
be
found
in
Exercise
6.3.21.
Adjoint
functor
theorems
Every
functor
with
a
left
adjoint
preserves
limits,
but
limit-preservation
alone
does
not
guarantee
the
existence
of
a
left
adjoint.
For
example,
let
B
be
any
category.
The
unique
functor
B
→
1
always
preserves
limits,
but
by
Exam-
ple
2.1.9,
it
only
has
a
left
adjoint
if
B
has
an
initial
object.
On
the
other
hand,
if
we
have
a
limit-preserving
functor
G
:
B
→
A
and
B
has
all
limits,
then
there
is
an
excellent
chance
that
G
has
a
left
adjoint.
It
is
still
not
always
true,
but
counterexamples
are
harder
to
find.
For
instance
(taking
A
=
1
again),
can
you
find
a
category
B
that
has
all
limits
but
no
initial
object?
The
condition
of
having
all
limits
is
so
important
that
it
has
its
own
word:
Definition
6.3.6
has
all
limits.
A
category
is
complete
(or
properly,
small
complete)
if
it
There
are
various
results
called
adjoint
functor
theorems,
all
of
the
following
form:
Let
A
be
a
category,
B
a
complete
category,
and
G
:
B
→
A
a
functor.
Suppose
that
A
,
B
and
G
satisfy
certain
further
conditions.
Then
G
has
a
left
adjoint
⇐⇒
G
preserves
limits.
The
forwards
implication
is
immediate
from
Theorem
6.3.1.
It
is
the
back-
wards
implication
that
concerns
us
here.
Typically,
the
‘further
conditions’
involve
the
distinction
between
small
and
6.3
Interactions
between
adjoint
functors
and
limits
161
large
collections.
But
there
is
a
special
case
in
which
these
complications
dis-
appear,
and
I
will
use
it
to
explain
the
main
idea
behind
the
proofs
of
the
adjoint
functor
theorems.
It
is
the
case
where
the
categories
A
and
B
are
ordered
sets.
As
we
saw
in
Section
5.1,
limits
in
ordered
sets
are
meets.
More
precisely,
if
D
:
I
→
B
is
a
diagram
in
an
ordered
set
B,
then
^
lim
D
=
D(I),
←I
I∈I
with
one
side
defined
if
and
only
if
the
other
is.
So
an
ordered
set
is
complete
if
and
only
if
every
subset
has
a
meet.
Similarly,
a
map
G
:
B
→
A
of
ordered
sets
preserves
limits
if
and
only
if
^
!
^
G
B
i
=
G(B
i
)
i∈I
i∈I
whenever
(B
i
)
i∈I
is
a
family
of
elements
of
B
for
which
a
meet
exists.
We
now
show
that
for
ordered
sets,
there
is
an
adjoint
functor
theorem
of
the
simplest
possible
kind:
there
are
no
‘further
conditions’
at
all.
Proposition
6.3.7
(Adjoint
functor
theorem
for
ordered
sets)
Let
A
be
an
ordered
set,
B
a
complete
ordered
set,
and
G
:
B
→
A
an
order-preserving
map.
Then
G
has
a
left
adjoint
⇐⇒
G
preserves
meets.
Proof
Suppose
that
G
preserves
meets.
By
Corollary
2.3.7,
it
is
enough
to
show
that
for
each
A
∈
A,
the
comma
category
(A
⇒
G)
has
an
initial
object.
Let
A
∈
A.
Then
(A
⇒
G)
is
an
ordered
set,
namely,
{B
∈
B
|
A
≤
G(B)}
with
the
order
inherited
from
B.
We
have
to
show
that
(A
⇒
G)
has
a
least
element.
V
Since
B
is
complete,
the
meet
B∈B
:
A≤G(B)
B
exists
in
B.
This
is
the
meet
of
all
the
elements
of
(A
⇒
G),
so
it
suffices
to
show
that
the
meet
is
itself
an
element
of
(A
⇒
G).
And
indeed,
since
G
preserves
meets,
we
have
!
^
^
G
B
=
G(B)
≥
A,
B∈B
:
A≤G(B)
B∈B
:
A≤G(B)
as
required.
In
the
general
setting
of
Corollary
2.3.7,
the
initial
object
of
(A
⇒
G)
is
the
η
A
pair
F(A),
A
−→
GF(A)
,
where
F
is
the
left
adjoint
and
η
is
the
unit
map.
So
in
Proposition
6.3.7,
the
left
adjoint
F
is
given
by
^
F(A)
=
B.
(6.19)
B∈B
:
A≤G(B)
162
Adjoints,
representables
and
limits
Example
6.3.8
Consider
Proposition
6.3.7
in
the
case
A
=
1.
The
unique
functor
G
:
B
→
1
automatically
preserves
meets,
and,
as
observed
above,
a
left
adjoint
to
G
is
an
initial
object
of
B.
So
in
the
case
A
=
1,
the
proposition
states
that
a
complete
ordered
set
has
a
least
element.
This
is
not
quite
trivial,
since
completeness
means
the
existence
of
all
meets,
whereas
a
least
element
is
an
empty
join.
V
By
(6.19),
the
least
element
of
B
is
B∈B
B.
Thus,
a
least
element
is
not
only
a
colimit
of
the
functor
∅
→
B;
it
is
also
a
limit
of
the
identity
functor
B
→
B.
The
synonym
‘least
upper
bound’
for
‘join’
suggests
a
theorem:
that
a
poset
with
all
meets
also
has
all
joins.
Indeed,
given
a
poset
B
with
all
meets,
the
join
of
a
subset
of
B
is
simply
the
meet
of
its
upper
bounds:
quite
literally,
its
least
upper
bound.
Let
us
now
attempt
to
extend
Proposition
6.3.7
from
ordered
sets
to
cate-
gories,
starting
with
a
limit-preserving
functor
G
from
a
complete
category
B
to
a
category
A
.
In
the
case
of
ordered
sets,
we
had
for
each
A
∈
A
an
inclu-
sion
map
P
A
:
(A
⇒
G)
,→
B,
and
we
showed
that
the
left
adjoint
F
was
given
by
F(A)
=
lim
P
A
.
←(A⇒G)
(6.20)
In
the
general
case,
the
analogue
of
the
inclusion
functor
is
the
projection
func-
tor
P
A
:
(A
⇒
G)
→
B
(6.21)
f
B,
A
−→
G(B)
7→
B.
The
case
of
ordered
sets
suggests
that
in
general,
equation
(6.20)
might
define
a
left
adjoint
F
to
G.
And
indeed,
it
can
be
shown
that
if
this
limit
in
B
exists
and
is
preserved
by
G,
then
(6.20)
really
does
give
a
left
adjoint
(Theorem
X.1.2
of
Mac
Lane
(1971)).
This
might
seem
to
suggest
that
our
adjoint
functor
theorem
generalizes
smoothly
from
ordered
sets
to
arbitrary
categories,
with
no
need
for
further
conditions.
But
it
does
not,
for
reasons
that
are
quite
subtle.
Those
reasons
are
more
easily
explained
if
we
relax
our
terminology
slightly.
When
we
defined
limits,
we
built
in
the
condition
that
the
shape
category
I
was
small.
However,
the
definition
of
limit
makes
sense
for
an
arbitrary
category
I.
In
this
discussion,
we
will
need
to
refer
to
this
more
inclusive
notion
of
limit,
so
let
us
temporarily
suspend
the
convention
that
the
shape
categories
I
of
limits
are
always
small.
Now,
in
the
template
for
adjoint
functor
theorems
stated
above
(after
Defini-
tion
6.3.6),
it
was
only
required
that
B
has,
and
G
preserves,
small
limits.
But
6.3
Interactions
between
adjoint
functors
and
limits
163
if
B
is
a
large
category
then
(A
⇒
G)
might
also
be
large,
since
to
specify
an
object
or
map
in
(A
⇒
G),
we
have
to
specify
(among
other
things)
an
object
or
map
in
B.
So,
the
limit
(6.20)
defining
the
left
adjoint
is
not
guaranteed
to
be
small.
Hence
there
is
no
guarantee
that
this
limit
exists
in
B,
nor
that
it
is
preserved
by
G.
It
follows
that
the
functor
F
‘defined’
by
(6.20)
might
not
be
defined
at
all,
let
alone
a
left
adjoint.
(The
reader
experiencing
difficulty
with
reasoning
about
small
and
large
collections
might
usefully
compare
finite
and
infinite
collections.
For
instance,
if
B
is
a
finite
category
and
A
has
finite
hom-sets
then
(A
⇒
G)
is
also
finite,
but
otherwise
(A
⇒
G)
might
be
infinite.)
Proposition
6.3.7
still
stands,
since
there
we
were
dealing
with
ordered
sets,
which
as
categories
are
small.
We
might
hope
to
extend
it
from
posets
to
ar-
bitrary
small
categories,
since
the
problem
just
described
affects
only
large
categories.
But
this
turns
out
not
to
be
very
fruitful,
since
in
fact,
complete
posets
are
the
only
complete
small
categories
(Exercise
6.3.23).
Alternatively,
we
could
try
to
salvage
the
argument
by
assuming
that
B
has,
and
G
preserves,
all
(possibly
large)
limits.
But
again,
this
is
unhelpful:
there
are
almost
no
such
categories
B.
The
situation
therefore
becomes
more
complicated.
Each
of
the
best-known
adjoint
functor
theorems
imposes
further
conditions
implying
that
the
large
limit
lim
P
A
can
be
replaced
by
a
small
limit
in
some
clever
way.
This
←(A⇒G)
allows
one
to
proceed
with
the
argument
above.
The
two
most
famous
adjoint
functor
theorems
are
the
‘general’
and
the
‘special’.
Their
exact
statements
and
proofs
are
perhaps
less
significant
than
their
consequences.
Definition
6.3.9
Let
C
be
a
category.
A
weakly
initial
set
in
C
is
a
set
S
of
objects
with
the
property
that
for
each
C
∈
C
,
there
exist
an
element
S
∈
S
and
a
map
S
→
C.
Note
that
S
must
be
a
set,
that
is,
small.
So,
the
existence
of
a
weakly
initial
set
is
some
kind
of
size
restriction.
Such
size
restrictions
are
comparable
to
finiteness
conditions
in
algebra.
Theorem
6.3.10
(General
adjoint
functor
theorem)
Let
A
be
a
category,
B
a
complete
category,
and
G
:
B
→
A
a
functor.
Suppose
that
B
is
locally
small
and
that
for
each
A
∈
A
,
the
category
(A
⇒
G)
has
a
weakly
initial
set.
Then
G
has
a
left
adjoint
⇐⇒
G
preserves
limits.
Proof
See
the
appendix.
164
Adjoints,
representables
and
limits
Example
6.3.11
The
general
adjoint
functor
theorem
(GAFT)
implies
that
for
any
category
B
of
algebras
(Grp,
Vect
k
,
.
.
.
),
the
forgetful
functor
U
:
B
→
Set
has
a
left
adjoint.
Indeed,
we
saw
in
Example
5.1.23
that
B
has
all
limits,
and
in
Example
5.3.4
that
U
preserves
them.
Also,
B
is
locally
small.
To
apply
GAFT,
we
now
just
have
to
check
that
for
each
A
∈
Set,
the
comma
category
(A
⇒
U)
has
a
weakly
initial
set.
This
requires
a
little
cardinal
arithmetic,
omitted
here;
see
Exercise
6.3.24.
So
GAFT
tells
us
that,
for
instance,
the
free
group
functor
exists.
In
Ex-
amples
1.2.4(a)
and
2.1.3(b),
we
began
to
see
the
trickiness
of
explicitly
con-
structing
the
free
group
on
a
generating
set
A.
One
has
to
define
the
set
of
‘formal
expressions’
(such
as
x
−1
yx
2
zy
−3
,
with
x,
y,
z
∈
A),
then
say
what
it
means
for
two
such
expressions
to
be
equivalent
(so
that
x
−2
x
5
y
is
equivalent
to
x
3
y),
then
define
F(A)
to
be
the
set
of
all
equivalence
classes,
then
define
the
group
structure,
then
check
the
group
axioms,
then
prove
that
the
resulting
group
has
the
universal
property
required.
But
using
GAFT,
we
can
avoid
these
complications
entirely.
The
price
to
be
paid
is
that
GAFT
does
not
give
us
an
explicit
description
of
free
groups
(or
left
adjoints
more
generally).
When
people
speak
of
knowing
some
object
‘explicitly’,
they
usually
mean
knowing
its
elements.
An
element
of
an
object
is
a
map
into
it,
and
we
have
no
handle
on
maps
into
F(A):
since
F
is
a
left
adjoint,
it
is
maps
out
of
F(A)
that
we
know
about.
This
is
why
explicit
descriptions
of
left
adjoints
are
often
hard
to
come
by.
Example
6.3.12
More
generally,
GAFT
guarantees
that
forgetful
functors
between
categories
of
algebras,
such
as
Ab
→
Grp,
Grp
→
Mon,
Ring
→
Mon,
Vect
C
→
Vect
R
,
have
left
adjoints.
(Some
of
them
are
described
in
Examples
2.1.3.)
This
is
‘more
generally’
because
Set
can
be
seen
as
a
degenerate
example
of
a
category
of
algebras,
in
the
sense
of
Remark
2.1.4:
a
group,
ring,
etc.,
is
a
set
equipped
with
some
operations
satisfying
some
equations,
and
a
set
is
a
set
equipped
with
no
operations
satisfying
no
equations.
The
special
adjoint
functor
theorem
(SAFT)
operates
under
much
tighter
hypotheses
than
GAFT,
and
is
much
less
widely
applicable.
Its
main
advantage
is
that
it
removes
the
condition
on
weakly
initial
sets.
Indeed,
it
removes
all
further
conditions
on
the
functor
G.
Theorem
6.3.13
(Special
adjoint
functor
theorem)
Let
A
be
a
category,
B
a
complete
category,
and
G
:
B
→
A
a
functor.
Suppose
that
A
and
B
6.3
Interactions
between
adjoint
functors
and
limits
165
are
locally
small,
and
that
B
satisfies
certain
further
conditions.
Then
G
has
a
left
adjoint
⇐⇒
G
preserves
limits.
A
precise
statement
and
proof
can
be
found
in
Section
V.8
of
Mac
Lane
(1971).
Example
6.3.14
Here
is
the
classic
application
of
SAFT.
Let
CptHff
be
the
category
of
compact
Hausdorff
spaces,
and
U
:
CptHff
→
Top
the
forgetful
functor.
SAFT
tells
us
that
U
has
a
left
adjoint
F,
turning
any
space
into
a
compact
Hausdorff
space
in
a
canonical
way.
The
existence
of
this
left
adjoint
is
far
from
obvious,
and
verifying
the
hy-
potheses
of
SAFT
(or
indeed,
constructing
F
in
any
other
way)
requires
some
deep
theorems
of
topology.
Given
a
space
X,
the
resulting
compact
Hausdorff
space
F(X)
is
called
its
Stone–Čech
compactification.
Provided
that
X
sat-
isfies
some
mild
separation
conditions,
the
unit
of
the
adjunction
at
X
is
an
embedding,
so
that
UF(X)
contains
X
as
a
subspace.
Another
advantage
of
SAFT
is
that
one
can
extract
from
its
proof
a
fairly
explicit
formula
for
the
left
adjoint.
In
this
case,
it
tells
us
that
F(X)
is
the
closure
of
the
image
of
the
canonical
map
X
→
[0,
1]
Top(X,[0,1])
,
where
the
codomain
is
a
power
of
[0,
1]
in
Top.
Cartesian
closed
categories
We
have
seen
that
for
every
set
B,
there
is
an
adjunction
(−
×
B)
a
(−)
B
(Ex-
ample
2.1.6),
and
that
for
every
category
B,
there
is
an
adjunction
(−
×
B)
a
[B,
−]
(Remark
4.1.23(c)).
Definition
6.3.15
A
category
A
is
cartesian
closed
if
it
has
finite
products
and
for
each
B
∈
A
,
the
functor
−
×
B
:
A
→
A
has
a
right
adjoint.
We
write
the
right
adjoint
as
(−)
B
,
and,
for
C
∈
A
,
call
C
B
an
exponential.
We
may
think
of
C
B
as
the
space
of
maps
from
B
to
C.
Adjointness
says
that
for
all
A,
B,
C
∈
A
,
A
(A
×
B,
C)
A
A,
C
B
naturally
in
A
and
C.
In
fact,
the
isomorphism
is
natural
in
B
too;
that
comes
for
free.
Example
6.3.16
Set
is
cartesian
closed;
C
B
is
the
function
set
Set(B,
C).
Example
6.3.17
CAT
is
cartesian
closed;
C
B
is
the
functor
category
[B,
C
].
166
Adjoints,
representables
and
limits
In
any
cartesian
closed
category
with
finite
sums,
the
isomorphisms
(6.17)
and
(6.18)
of
Example
6.3.3
hold,
for
the
same
reasons
as
stated
there.
The
objects
of
a
cartesian
closed
category
therefore
possess
an
arithmetic
like
that
of
the
natural
numbers.
This
thought
can
be
developed
in
several
interesting
directions,
but
here
we
just
note
that
these
isomorphisms
provide
a
way
of
proving
that
a
category
is
not
cartesian
closed.
Example
6.3.18
Vect
k
is
not
cartesian
closed,
for
any
field
k.
It
does
have
finite
products,
as
we
saw
in
Example
5.1.5:
binary
product
is
direct
sum
⊕,
and
the
terminal
object
is
the
trivial
vector
space
{0},
which
is
also
initial.
But
if
Vect
k
were
cartesian
closed
then
equations
(6.17)
would
hold,
so
that
{0}
⊕
B
{0}
for
all
vector
spaces
B.
This
is
plainly
false.
Remark
6.3.19
For
any
vector
spaces
V
and
W,
the
set
Vect
k
(V,
W)
of
linear
maps
can
itself
be
given
the
structure
of
a
vector
space,
as
in
Example
1.2.12.
Let
us
now
call
this
vector
space
[V,
W].
Given
that
exponentials
are
supposed
to
be
‘spaces
of
maps’,
you
might
expect
Vect
k
to
be
cartesian
closed,
with
[−,
−]
as
its
exponential.
We
have
just
seen
that
this
cannot
be
so.
But
as
it
turns
out,
the
linear
maps
U
→
[V,
W]
correspond
to
the
bilinear
maps
U
×
V
→
W,
or
equivalently
the
linear
maps
U⊗V
→
W.
In
the
jargon,
Vect
k
is
an
example
of
a
‘monoidal
closed
category’.
These
are
like
cartesian
closed
categories,
but
with
the
cartesian
(categorical)
product
replaced
by
some
other
operation
called
‘product’,
in
this
case
the
tensor
product
of
vector
spaces.
For
any
set
I,
the
product
category
Set
I
is
cartesian
closed,
just
because
Set
is.
(Exponentials
in
Set
I
,
as
well
as
products,
are
computed
pointwise.)
Put
an-
other
way,
[A
op
,
Set]
is
cartesian
closed
whenever
A
is
discrete.
We
now
show
that,
in
fact,
[A
op
,
Set]
is
cartesian
closed
for
any
small
category
A
whatsoever.
In
preparation
for
proving
this,
let
us
conduct
a
thought
experiment.
Write
Â
=
[A
op
,
Set].
If
Â
is
cartesian
closed,
what
must
exponentials
in
Â
be?
In
other
words,
given
presheaves
Y
and
Z,
what
must
Z
Y
be
in
order
that
Â
X,
Z
Y
Â(X
×
Y,
Z)
(6.22)
for
all
presheaves
X?
If
this
is
true
for
all
presheaves
X,
then
in
particular
it
is
true
when
X
is
representable,
so
Z
Y
(A)
Â
H
A
,
Z
Y
Â(H
A
×
Y,
Z)
for
all
A
∈
A,
the
first
step
by
Yoneda.
This
tells
us
what
Z
Y
must
be.
Notice
that
Z
Y
(A)
is
not
simply
Z(A)
Y(A)
,
as
one
might
at
first
guess:
exponentials
in
a
presheaf
category
are
not
generally
computed
pointwise.
6.3
Interactions
between
adjoint
functors
and
limits
Theorem
6.3.20
sian
closed.
167
For
any
small
category
A,
the
presheaf
category
Â
is
carte-
Here
is
the
strategy
of
the
proof.
The
argument
in
the
thought
experiment
gives
us
the
isomorphism
(6.22)
whenever
X
is
representable.
A
general
presheaf
X
is
not
representable,
but
it
is
a
colimit
of
representables,
and
this
allows
us
to
bootstrap
our
way
up.
Proof
We
know
that
Â
has
all
limits,
and
in
particular,
finite
products.
It
remains
to
show
that
Â
has
exponentials.
Fix
Y
∈
Â.
First
we
prove
that
−
×
Y
:
Â
→
Â
preserves
colimits.
(Eventually
we
will
prove
that
−
×
Y
has
a
right
adjoint,
from
which
preservation
of
colimits
fol-
lows,
but
our
proof
that
it
has
a
right
adjoint
will
use
preservation
of
colimits.)
Indeed,
since
products
and
colimits
in
Â
are
computed
pointwise,
it
is
enough
to
prove
that
for
any
set
S
,
the
functor
−
×
S
:
Set
→
Set
preserves
colimits,
and
this
follows
from
the
fact
that
Set
is
cartesian
closed.
For
each
presheaf
Z
on
A,
let
Z
Y
be
the
presheaf
defined
by
Z
Y
(A)
=
Â(H
A
×
Y,
Z)
for
all
A
∈
A.
This
defines
a
functor
(−)
Y
:
Â
→
Â.
I
claim
that
(−
×
Y)
a
(−)
Y
.
Let
X,
Z
∈
Â.
Write
P
:
E(X)
→
A
for
the
projection
(as
in
Definition
6.2.16),
and
write
H
P
=
H
•
◦
P.
Then
Â
X,
Z
Y
Â
lim
H
P
,
Z
Y
(6.23)
→E(X)
lim
op
Â
H
P
,
Z
Y
(6.24)
←E(X)
lim
op
Z
Y
(P)
(6.25)
lim
op
Â(H
P
×
Y,
Z)
←E(X)
Â
lim
(H
P
×
Y),
Z
→E(X)
Â
lim
H
P
×
Y,
Z
(6.26)
Â(X
×
Y,
Z)
(6.29)
←E(X)
→E(X)
(6.27)
(6.28)
naturally
in
X
and
Z.
Here
(6.23)
and
(6.29)
follow
from
Theorem
6.2.17;
(6.24)
and
(6.27)
are
because
representables
preserve
limits
(as
rephrased
in
Remark
6.2.3);
(6.25)
is
by
Yoneda;
(6.26)
is
by
definition
of
Z
Y
;
and
(6.28)
is
because
−
×
Y
preserves
colimits.
168
Adjoints,
representables
and
limits
This
result
can
be
seen
as
a
step
along
the
road
to
topos
theory.
A
topos
is
a
category
with
certain
special
properties.
Topos
theory
unifies,
in
an
extraordi-
nary
way,
important
aspects
of
logic
and
geometry.
For
instance,
a
topos
can
be
regarded
as
a
‘universe
of
sets’:
Set
is
the
most
basic
example
of
a
topos,
and
every
topos
shares
enough
features
with
Set
that
one
can
reason
with
its
objects
as
if
they
were
sets
of
some
exotic
kind.
On
the
other
hand,
a
topos
can
be
regarded
as
a
generalized
topological
space:
every
space
gives
rise
to
a
topos
(namely,
the
category
of
sheaves
on
it),
and
topolog-
ical
properties
of
the
space
can
be
reinterpreted
in
a
useful
way
as
categorical
properties
of
its
associated
topos.
By
definition,
a
topos
is
a
cartesian
closed
category
with
finite
limits
and
with
one
further
property:
the
existence
of
a
so-called
subobject
classifier.
For
example,
the
two-element
set
2
is
the
subobject
classifier
of
Set,
which
means,
informally,
that
subsets
of
a
set
A
correspond
one-to-one
with
maps
A
→
2.
Exercises
6.3.26
and
6.3.27
give
the
formal
definition
of
subobject
classifier,
then
guide
you
through
the
proof
that
Set,
and,
more
generally,
every
presheaf
category,
is
a
topos.
Exercises
6.3.21
(a)
Prove
that
the
forgetful
functor
U
:
Grp
→
Set
has
no
right
ad-
joint.
(b)
Prove
that
the
chain
of
adjunctions
C
a
D
a
O
a
I
in
Exercise
3.2.16
extends
no
further
in
either
direction.
(c)
Does
the
chain
of
adjunctions
in
Exercise
2.1.17
extend
further
in
either
direction?
6.3.22
Let
A
be
a
locally
small
category.
For
functors
U
:
A
→
Set,
con-
sider
the
following
three
conditions:
(A)
U
has
a
left
adjoint;
(R)
U
is
repre-
sentable;
(L)
U
preserves
limits.
(a)
Show
that
(A)
=⇒
(R)
=⇒
(L).
(b)
Show
that
if
A
has
sums
then
(R)
=⇒
(A).
(If
A
satisfies
the
hypotheses
of
the
special
adjoint
functor
theorem
then
also
(L)
=⇒
(A),
so
the
three
conditions
are
equivalent.)
6.3.23
(a)
Prove
that
every
preordered
set
is
equivalent
(as
a
category)
to
an
ordered
set.
(b)
Let
A
be
a
category
with
all
small
products.
Suppose
that
A
is
not
a
f
preorder,
so
that
there
exists
a
parallel
pair
of
maps
A
g
/
/
B
in
A
with
6.3
Interactions
between
adjoint
functors
and
limits
169
f
,
g.
By
considering
the
maps
A
→
B
I
for
each
set
I,
prove
that
A
is
not
small.
(c)
Deduce
that
every
small
category
with
small
products
is
equivalent
to
a
complete
ordered
set.
(d)
Adapt
the
argument
to
prove
that
every
finite
category
with
finite
products
is
equivalent
to
a
complete
ordered
set.
6.3.24
Probably
the
most
important
application
of
the
general
adjoint
functor
theorem
is
to
proving
that
forgetful
functors
between
categories
of
algebras
have
left
adjoints
(Example
6.3.11).
Verifying
the
hypotheses
can
be
done
with
some
cardinal
arithmetic.
Here
is
a
typical
example.
(a)
Let
A
be
a
set.
Prove
that
for
any
group
G
and
family
(g
a
)
a∈A
of
elements
of
G,
the
subgroup
of
G
generated
by
{g
a
|
a
∈
A}
has
cardinality
at
most
max{|N|
,
|A|}.
(b)
Prove
that
for
any
set
S
,
the
collection
of
isomorphism
classes
of
groups
of
cardinality
at
most
|S
|
is
small.
(c)
Let
U
:
Grp
→
Set
be
the
forgetful
functor
from
groups
to
sets.
Deduce
from
(a)
and
(b)
that
for
every
set
A,
the
comma
category
(A
⇒
U)
has
a
weakly
initial
set.
(d)
Use
GAFT
to
conclude
that
U
has
a
left
adjoint.
6.3.25
Let
A
be
a
small
cartesian
closed
category.
Prove
that
the
Yoneda
embedding
A
→
[A
op
,
Set]
preserves
the
whole
cartesian
closed
structure
(ex-
ponentials
as
well
as
products).
6.3.26
Recall
from
Exercise
5.1.40
the
notion
of
subobject.
A
category
A
is
well-powered
if
for
each
A
∈
A
,
the
class
of
subobjects
of
A
is
small,
that
is,
a
set.
(All
of
our
usual
examples
of
categories
are
well-powered.)
Let
A
be
a
well-powered
category
with
pullbacks,
and
write
Sub(A)
for
the
set
of
subobjects
of
an
object
A
∈
A
.
f
(a)
Deduce
from
Exercise
5.1.42
that
any
map
A
0
−→
A
in
A
induces
a
map
Sub(
f
)
:
Sub(A)
→
Sub(A
0
).
(b)
Show
that
this
determines
a
functor
Sub
:
A
op
→
Set.
(Hint:
use
Exer-
cise
5.1.35.)
(c)
For
some
categories
A
,
the
functor
Sub
is
representable.
A
subobject
classifier
for
A
is
an
object
Ω
∈
A
such
that
Sub
H
Ω
.
Prove
that
2
is
a
subobject
classifier
for
Set.
A
topos
is
a
cartesian
closed
category
with
finite
limits
and
a
subobject
classi-
fier.
You
have
just
completed
the
proof
that
Set
is
a
topos.
170
Adjoints,
representables
and
limits
6.3.27
This
exercise
follows
on
from
the
last,
culminating
in
the
proof
that
every
presheaf
category
is
a
topos.
Let
A
be
a
small
category.
(a)
By
conducting
a
thought
experiment
similar
to
the
one
before
the
statement
of
Theorem
6.3.20,
find
out
what
the
subobject
classifier
Ω
of
[A
op
,
Set]
must
be
if
it
exists.
(b)
Prove
that
this
Ω
is
indeed
a
subobject
classifier.
(c)
Conclude
that
[A
op
,
Set]
is
a
topos.
Appendix
Proof
of
the
general
adjoint
functor
theorem
Here
we
prove
the
general
adjoint
functor
theorem,
which
for
convenience
is
restated
below.
The
left-to-right
implication
follows
immediately
from
Theo-
rem
6.3.1;
it
is
the
right-to-left
implication
that
we
have
to
prove.
Theorem
6.3.10
(General
adjoint
functor
theorem)
Let
A
be
a
category,
B
a
complete
category,
and
G
:
B
→
A
a
functor.
Suppose
that
B
is
locally
small
and
that
for
each
A
∈
A
,
the
category
(A
⇒
G)
has
a
weakly
initial
set.
Then
G
has
a
left
adjoint
⇐⇒
G
preserves
limits.
The
heart
of
the
proof
is
the
case
A
=
1,
where
GAFT
asserts
that
a
com-
plete
locally
small
category
with
a
weakly
initial
set
has
an
initial
object.
We
prove
this
first.
The
proof
of
this
special
case
is
illuminated
by
considering
the
even
more
special
case
where
A
=
1
and
the
category
B
is
a
poset
B.
We
saw
in
Ex-
ample
6.3.8
that
the
initial
object
(least
element)
of
a
complete
poset
B
can
be
constructed
as
the
meet
of
all
its
elements.
Otherwise
put,
it
is
the
limit
of
the
identity
functor
1
B
:
B
→
B.
One
might
try
to
extend
this
result
to
arbitrary
categories
B
by
proving
that
the
limit
of
the
identity
functor
1
B
:
B
→
B
is
(if
it
exists)
an
initial
object.
This
is
indeed
true
(Exercise
A.3
below).
However,
it
is
unhelpful:
for
if
B
is
large
then
the
limit
of
1
B
is
a
large
limit,
but
we
are
only
given
that
B
has
small
limits.
We
seem
to
be
at
an
impasse
–
but
this
is
where
the
clever
idea
behind
GAFT
comes
in.
In
order
to
construct
the
least
element
of
a
complete
poset,
it
is
not
necessary
to
take
the
meet
of
all
the
elements.
More
economically,
we
could
just
take
the
meet
of
the
elements
of
some
weakly
initial
subset
(Exercise
A.4).
171
172
Proof
of
the
general
adjoint
functor
theorem
In
general,
for
an
arbitrary
complete
category,
the
limit
of
any
weakly
initial
set
is
an
initial
object.
We
prove
this
now.
Lemma
A.1
Let
C
be
a
complete
locally
small
category
with
a
weakly
initial
set.
Then
C
has
an
initial
object.
Proof
Let
S
be
a
weakly
initial
set
in
C
.
Regard
S
as
a
full
subcategory
of
C
;
then
S
is
small,
since
C
is
locally
small.
We
may
therefore
take
a
limit
cone
p
S
0
−→
S
(A.1)
S
∈S
of
the
inclusion
S
,→
C
.
We
prove
that
0
is
initial.
Let
C
∈
C
.
We
have
to
show
that
there
is
exactly
one
map
0
→
C.
Certainly
there
is
at
least
one,
since
we
may
choose
some
S
∈
S
and
map
j
:
S
→
C,
and
we
then
have
the
composite
jp
S
:
0
→
C.
To
prove
uniqueness,
let
f,
g
:
0
→
C.
Form
the
equalizer
E
i
/
0
f
g
/
/
C
.
Since
S
is
weakly
initial,
we
may
choose
S
∈
S
and
h
:
S
→
E.
We
then
have
maps
0
p
S
/
S
h
/
E
i
/
0
with
the
property
that
for
all
S
0
∈
S,
p
S
0
(ihp
S
)
=
(p
S
0
ih)p
S
=
p
S
0
=
p
S
0
1
0
(where
the
second
equality
follows
from
(A.1)
being
a
cone).
But
(A.1)
is
a
limit
cone,
so
ihp
S
=
1
0
by
Exercise
5.1.36(a).
Hence
f
=
f
ihp
S
=
gihp
S
=
g,
as
required.
We
have
now
proved
GAFT
in
the
special
case
A
=
1.
The
rest
of
the
proof
is
comparatively
routine.
Lemma
A.2
Let
A
and
B
be
categories.
Let
G
:
B
→
A
be
a
functor
that
preserves
limits.
Then
the
projection
functor
P
A
:
(A
⇒
G)
→
B
of
(6.21)
creates
limits,
for
each
A
∈
A
.
In
particular,
if
B
is
complete
then
so
is
each
comma
category
(A
⇒
G).
Proof
The
first
statement
is
Exercise
A.5(b),
and
the
second
follows
from
Lemma
5.3.6.
Proof
of
the
general
adjoint
functor
theorem
173
We
now
prove
GAFT.
By
Corollary
2.3.7,
it
is
enough
to
show
that
(A
⇒
G)
has
an
initial
object
for
each
A
∈
A
.
Let
A
∈
A
.
By
Lemma
A.2,
(A
⇒
G)
is
complete,
and
by
hypothesis,
it
has
a
weakly
initial
set.
It
is
also
locally
small,
since
B
is.
Hence
by
Lemma
A.1,
it
has
an
initial
object,
as
required.
Exercises
A.3
In
this
exercise,
we
suspend
the
convention
(made
implicitly
in
Defini-
tion
5.1.19)
that
we
only
speak
of
the
limit
of
a
functor
I
→
C
when
I
is
small.
Let
B
be
a
category,
possibly
large.
The
aim
is
to
prove
that
a
limit
of
the
identity
functor
on
B
is
exactly
an
initial
object
of
B.
(a)
Let
0
be
an
initial
object
of
B.
Show
that
the
cone
(0
→
B)
B∈B
on
the
identity
functor
1
B
is
a
limit
cone.
p
B
(b)
Now
let
L
−→
B
be
a
limit
cone
on
1
B
.
Prove
that
p
L
is
the
identity
B∈B
on
L,
and
deduce
that
L
is
initial.
A.4
Here
you
will
prove
the
special
case
of
Lemma
A.1
in
which
the
category
concerned
is
a
poset.
Let
C
be
a
poset
and
S
⊆
C.
(a)
What
does
it
mean,
in
purely
order-theoretic
terms,
for
S
to
be
a
weakly
initial
set
in
C?
V
(b)
Prove
directly
that
if
S
is
weakly
initial
and
the
meet
s∈S
s
exists
then
V
s∈S
s
is
a
least
element
of
C.
A.5
Let
G
:
B
→
A
be
a
limit-preserving
functor,
and
let
A
∈
A
.
(a)
Show
that
for
any
small
category
I,
a
diagram
of
shape
I
in
(A
⇒
G)
amounts
to
a
diagram
E
of
shape
I
in
B
together
with
a
cone
on
G
◦
E
with
vertex
A.
(b)
Prove
that
the
projection
functor
P
A
:
(A
⇒
G)
→
B
of
(6.21)
creates
limits.
Further
reading
This
book
is
intentionally
short.
Even
some
topics
that
are
included
in
most
introductions
to
category
theory
are
omitted
here.
I
will
indicate
some
of
the
topics
that
lie
beyond
the
scope
of
this
book,
and
suggest
where
you
might
read
about
them.
Since
there
is
far
more
written
on
category
theory
than
anyone
could
read
in
a
lifetime,
these
recommendations
are
necessarily
subjective.
The
towering
presence
among
category
theory
books
is
the
classic
by
one
of
its
founders:
Saunders
Mac
Lane,
Categories
for
the
Working
Mathematician.
Springer,
1971;
second
edition
with
two
new
chapters,
1998.
It
is
so
well-written
that
more
than
forty
years
on,
it
is
still
the
most
popular
in-
troduction
to
the
subject.
It
addresses
a
more
mature
readership
than
this
text,
and
covers
many
topics
omitted
here,
including
monads
(one
formalization
of
the
idea
of
algebraic
theory),
monoidal
categories
(categories
equipped
with
a
tensor
product),
2-categories
(mentioned
at
the
end
of
our
Chapter
1),
abelian
categories
(categories
of
modules),
ends
(an
elegant
generalization
of
the
no-
tion
of
limit),
and
Kan
extensions
(which
provide
the
tongue-in-cheek
title
of
the
book’s
final
section:
‘All
concepts
are
Kan
extensions’).
Another
well-liked
book,
longer
than
the
one
you
hold
in
your
hands
but
written
for
a
similar
readership,
is:
Steve
Awodey,
Category
Theory.
Oxford
University
Press,
2010.
Awodey’s
book
covers
less
than
Mac
Lane’s,
but
is
particularly
strong
on
con-
nections
between
category
theory
and
other
parts
of
logic.
It
has
a
full
chapter
on
cartesian
closed
categories,
and
also
covers
the
theory
of
monads.
Those
who
prefer
lectures
to
books
might
try
this
library
of
75
ten-minute
introductory
category
theory
videos:
174
Further
reading
175
Eugenia
Cheng
and
Simon
Willerton,
The
Catsters.
Available
at
https://www.youtube.com/user/TheCatsters,
2007–2010.
Other
than
the
topics
treated
here,
they
cover
monads,
enriched
categories,
in-
ternal
groups
(and
other
internal
algebraic
structures),
string
diagrams
(which
we
touched
on
in
Remark
2.2.9),
and
several
more
sophisticated
topics.
For
inspiration
as
much
as
instruction,
here
are
two
further
recommenda-
tions.
Saunders
Mac
Lane,
Mathematics:
Form
and
Function.
Springer,
1986.
F.
William
Lawvere
and
Stephen
H.
Schanuel,
Conceptual
Math-
ematics:
A
First
Introduction
to
Categories.
Cambridge
University
Press,
1997.
Mathematics:
Form
and
Function
is
a
tour
through
much
of
pure
and
applied
mathematics,
written
from
a
categorical
perspective.
Its
declared
purpose
is
to
present
the
author’s
philosophy
of
mathematics,
but
it
can
also
be
enjoyed
for
its
many
excellent
vignettes
of
exposition.
(Beware
of
the
numerous
small
errors.)
Conceptual
Mathematics
is
a
thought-provoking
text
and
an
intriguing
experiment:
category
theory
for
high-school
students,
complete
with
classroom
dialogues.
For
categorical
topics
beyond
the
scope
of
this
book,
two
good
general
ref-
erences
are:
Francis
Borceux,
Handbook
of
Categorical
Algebra,
Volumes
1–3.
Cambridge
University
Press,
1994.
Various
authors,
The
nLab.
Available
at
https://ncatlab.org,
2008–
present.
Borceux’s
encyclopaedic
work
often
takes
a
different
point
of
view
from
the
present
text,
but
covers
many,
many
more
topics.
Apart
from
those
just
men-
tioned
in
connection
with
other
books,
some
of
the
more
important
ones
are
fibrations,
bimodules
(also
called
profunctors
or
distributors),
Lawvere
theo-
ries,
Cauchy
completeness,
Morita
equivalence,
absolute
colimits,
and
flatness.
The
nLab
is
an
ever-growing
online
resource
for
mathematics,
focusing
on
category
theory
and
operating
on
similar
principles
to
Wikipedia.
Individual
entries
can
be
idiosyncratic,
but
it
has
become
a
very
useful
reference
for
ad-
vanced
categorical
topics.
Vigorous
research
in
category
theory
continues
to
be
done.
The
sources
listed
above
provide
ample
onward
references
for
anyone
wishing
to
explore.
176
Further
reading
Other
texts
cited
Timothy
Gowers,
Mathematics:
A
Very
Short
Introduction.
Oxford
University
Press,
2002.
G.
M.
Kelly,
Basic
Concepts
of
Enriched
Category
Theory.
Cam-
bridge
University
Press,
1982.
Also
Reprints
in
Theory
and
Appli-
cations
of
Categories
10
(2005),
1–136,
available
at
http://www.tac.
mta.ca/tac/reprints.
F.
William
Lawvere
and
Robert
Rosebrugh,
Sets
for
Mathematics.
Cambridge
University
Press,
2003.
Tom
Leinster,
Rethinking
set
theory.
American
Mathematical
Mon-
thly
121
(2014),
no.
5,
403–415.
Also
available
at
https://arxiv.org/
abs/1212.6543.
Index
of
notation
blank
space,
24
g
f
,
10
αF,
37
Fα,
37
α
A
,
28
A
(A,
B),
10
A
(A,
−),
84
A
(−,
A),
88
A
(
f,
−),
88
A
(−,
f
),
90
A
(A,
D),
147
D(−)(A),
149
B
A
,
30
B
A
,
69,
112,
165
(
f
i
)
i∈I
,
111
A,
B,
.
.
.
(typeface),
118
−,
24
¯,
42,
119,
127
˜,
96
ˆ,
96,
166
(
)
•
,
(
)
•
,
151
∗,
37
V
∗
,
24
f
∗
,
23,
88
f
∗
,
84,
90
◦,
10,
18,
30
g
◦
−,
84,
90
−
◦
f
,
88
∀,
3
∃!,
3
→,
10
,→,
6
∼
−→,
99
⇒,
29,
59,
60
a,
41
⊥,
>,
49
,
12,
26,
32
',
34
≤,
15,
74
|
|,
74
[
,
],
30
⊗,
6
×,
Q
16,
68,
109
,
68,
111
+,
P
68,
127
,
68,
127
q,
`
68
,
127
⊕,
110
A
/A,
59
A/A
,
60
A/∼,
70
∧,
V
111
,
111
∨,
W
128
,
128
(
)
−1
,
12
∅,
13
0,
127
1,
1,
10,
18,
30,
112
1,
13
2,
31,
69
2,
118
∆,
50,
73,
143
ε,
51
η,
51
π
1
,
21
χ,
69
Ab,
18
(
)
ab
,
45
Bilin,
86
C,
23
CAT,
18
Cat,
77
Cone,
143
CptHff,
123
177
CRing,
19
D,
4
E,
118,
155
ev,
149
FDVect,
32
Field,
46
FinSet,
35
Grp,
11
H
A
,
84
H
A
,
88
H
f
,
88
H
f
,
90
H
•
,
88
H
•
,
90
Hom,
10,
90
Hom,
23
I,
7
lim
,
119
←
lim
,
127
→
Mon,
18
N,
15
O,
24,
89
ob,
10
(
)
op
,
16
P,
118
P,
55,
69,
89
P
A
,
162
Ring,
11
S
1
,
85
Set,
11
T,
118
Top,
12
Top
∗
,
21
Toph,
17
Toph
∗
,
85
Vect
k
,
12
Z[x],
8
Index
abelianization,
45
adjoint
functor
theorems,
159–164
general,
162,
171–173
special,
163
adjunction,
41
composition
of
adjunctions,
49
vs.
equivalence,
55
fixed
points
of,
57
free–forgetful,
43–46
via
initial
objects,
60–63,
100,
101
limits
preserved
in,
158
naturality
axiom
for,
42,
50–51,
91,
101
nonexistence
of
adjoints,
159
uniqueness
of
adjoints,
43,
106
aerial
photography,
87
algebra,
92
for
algebraic
theory,
46
associative,
42–43
algebraic
geometry,
21,
36,
92
algebraic
theory,
46
algebraic
topology,
20
applied
mathematics,
9
arithmetic,
69,
112,
158,
165
cardinal,
163,
168
arity,
46
arrow,
10,
see
also
map
associative
algebra,
42–43
associativity,
10,
151
axiom
of
choice,
71,
135
bicycle
inner
tube,
133
bilinear,
see
map,
bilinear
black
king,
72
Boolean
algebra,
36
C
∗
-algebra,
36
canonical,
33,
39
Cantor,
Georg,
78
Cantor’s
theorem,
74
Cantor–Bernstein
theorem,
74
cardinality,
74,
163,
168
cartesian
closed
category,
164–167
category,
10
cartesian
closed,
164–167
category
of
categories,
18,
77
adjunctions
with
Set,
78,
167
comma,
see
comma
category
complete,
159
coslice,
60
discrete,
13,
78,
87
functor
out
of,
29,
31,
32
drawing
of,
13
of
elements,
154,
156
equivalence
of
categories,
34
vs.
adjunction,
55
essentially
small,
76
finite,
121
isomorphism
of
categories,
26
large,
75
locally
small,
75,
84
monoidal
closed,
165
one-object,
14–15,
see
also
monoid
and
group
opposite,
16
product
of
categories,
16,
26,
39
slice,
see
slice
category
slimmed-down,
35
small,
75,
118
2-category
of
categories,
38
well-powered,
168
centre,
26
characteristic
function,
69
chess,
72
178
Index
class,
11,
75
closure,
55
cocone,
126,
see
also
cone
codomain,
11
coequalizer,
128,
see
also
equalizer
cohomology,
24
colimit,
126,
see
also
limit
and
integration,
151
map
out
of,
147
collection,
11
comma
category,
59
limits
in,
172
commutes,
11
complete,
159
component
of
map
into
product,
111
of
natural
transformation,
28
composition,
10
horizontal,
37
vertical,
37
computer
science,
9,
79,
80
cone,
118
limit,
119
as
natural
transformation,
142
set
of
cones
as
limit,
146
connectedness,
156
contravariant,
22,
90
coproduct,
127,
see
also
sum
coprojection,
126
coreflective,
46
coslice
category,
60
counit,
see
unit
and
counit
covariant,
22
creation
of
limits,
138–139,
172
density,
154,
156
determinant,
29
diagonal,
see
functor,
diagonal
diagram,
118
commutative,
11
string,
55
direct
limit,
131
discrete,
see
category,
discrete
and
topological
space,
discrete
disjoint
union,
68,
see
also
set,
category
of,
sums
in
domain,
11
duality,
16,
35,
132
algebra–geometry,
23,
35
Gelfand–Naimark,
36
Pontryagin,
36
principle
of,
16,
49
Stone,
36
terminology
for,
126
for
vector
spaces,
24,
32
duck,
104
Eilenberg,
Samuel,
9
element
category
of
elements,
154,
156
as
function,
67
generalized,
92,
105,
117,
123,
156
least,
see
least
element
of
presheaf,
99
universal,
100
embedding,
102
empty
family,
111,
127
epic,
133,
see
also
monic
regular,
135
split,
135
epimorphism,
133,
see
also
epic
equalizer,
112,
132
map
into,
146
vs.
pullback,
124
of
sets,
70,
113
equivalence
of
categories,
34
vs.
adjunction,
55
equivalence
relation,
70,
135
generated
by
relation,
128
equivariant,
29
essentially
small,
76
essentially
surjective
on
objects,
34
evaluation,
32,
95,
148
explicit
description,
44,
163
exponential,
164,
see
also
set
of
functions
preserved
by
Yoneda
embedding,
168
faithful,
25,
27
family,
68
empty,
111,
127
fibred
product,
115,
see
also
pullback
field,
46,
83,
159
figure,
see
element,
generalized
fixed
point,
57,
77
forgetful,
see
functor,
forgetful
fork,
112
foundations,
71–73,
80
Fourier
analysis,
36,
78
free
functor,
19
Fubini’s
theorem,
151
full,
see
functor,
full
and
subcategory,
full
function
characteristic,
69
injective,
123
intuitive
description
of,
66
179
180
Index
number
of
functions,
67
partial,
64
set
of
functions,
47,
69,
164
surjective,
133
functor,
17
category,
30,
38,
164
limits
in,
148–153
composition
of
functors,
18
contravariant,
22,
90
covariant,
22
diagonal,
50,
73,
142
essentially
surjective
on
objects,
34
faithful,
25,
27
forgetful,
18
left
adjoint
to,
43,
87,
163
preserves
limits,
158
is
representable,
85,
87
free,
19
full,
25
full
and
faithful,
34,
103
identity,
18
limit
of,
171,
173
image
of,
25
product
of
functors,
148
representable,
84,
89
and
adjoints,
86,
167
colimit
of
representables,
153–156
isomorphism
of
representables,
104–105
limit
of
representables,
152–153
preserves
limits,
145–147
sum
of
representables,
156
‘seeing’,
83,
85
set-valued,
84
G-set,
22,
50,
157,
see
also
monoid,
action
of
general
adjoint
functor
theorem
(GAFT),
162,
171–173
generalized
element,
see
element,
generalized
generated
equivalence
relation,
128
greatest
common
divisor,
110
greatest
lower
bound,
111
group,
6,
101,
103,
see
also
monoid
abelian
coequalizer
of,
130
finite
limit
of,
123
abelianization
of,
45
action
of,
50,
157,
see
also
monoid,
action
of
category
of
groups,
11
colimits
in,
137
epics
in,
134
equalizers
in,
114
is
not
essentially
small,
77
isomorphisms
in,
12
limits
in,
121,
137–140
is
locally
small,
76
monics
in,
123
free,
19,
44,
63,
163,
168
free
on
monoid,
45
fundamental,
7,
21,
85,
131
isomorphism
of
elements
of,
39
non-homomorphisms
of
groups,
36
normal
subgroup
of,
135
as
one-object
category,
14
opposite,
26
order
of
element
of,
85,
105
representation
of,
see
representation
topological,
36
holomorphic
function,
153
hom-set,
75,
90
homology,
21
homotopy,
17,
85,
see
also
group,
fundamental
identity,
10
as
zero-fold
composite,
11
image
of
functor,
25
of
homomorphism,
130
inverse,
see
inverse
image
inclusion,
6
indiscrete
space,
7,
47
infimum,
111
∞-category,
38
initial,
see
object,
initial
and
set,
weakly
initial
injection,
123
injective
object,
140
integers,
see
Z
interchange
law,
38
intersection,
110,
120
as
pullback,
116,
130
inverse,
12
image,
57,
89
as
pullback,
115
limit,
120
right,
71
isomorphism,
12
of
categories,
26
and
full
and
faithful
functors,
103
natural,
31
preserved
by
functors,
26
join,
128
Kan
extension,
157
kernel,
6,
8,
114
Index
Kronecker,
Leopold,
78
large,
75
least
element,
128,
171,
173
as
meet,
161
least
upper
bound,
128
Lie
algebra,
42–43
limit,
118
as
adjoint,
144
vs.
colimit,
132,
147,
161
non-commutativity
with
colimits,
152
commutativity
with
limits,
150,
159
computed
pointwise,
148
cone,
119
creation
of,
138–139,
172
direct,
131
finite,
121
in
functor
category,
148–153
functoriality
of,
139
has
limits,
121
of
identity,
171,
173
informal
usage,
119
inverse,
120
large,
161–162,
171,
173
map
between
limits,
143
map
into,
147
non-pointwise,
150
preservation
of,
136
by
adjoint,
158
from
products
and
equalizers,
121
from
pullbacks
and
terminal
object,
125
reflection
of,
136
as
representation
of
cone
functor,
142
small,
119,
161–162,
173
uniqueness
of,
143,
145
locally
small,
75,
84
loop,
92
lower
bound,
111
lowest
common
multiple,
128
Mac
Lane,
Saunders,
9
manifold,
133
map,
10
bilinear,
4,
86,
105,
165
need
not
resemble
function,
13
order-preserving,
22,
26
matrix,
40
meet,
111
metric
space,
91
minimum,
110
model,
46
monic,
123
composition
of
monics,
135
181
pullback
of,
125,
135
regular,
135
split,
135
monoid,
15
action
of,
22,
24,
29,
31,
85,
see
also
group,
action
of
epics
between
monoids,
134
free
group
on,
45
homomorphism
of
monoids,
21
as
one-object
category,
15,
29,
35,
77
opposite,
26
Yoneda
lemma
for
monoids,
99
monoidal
closed
category,
165
monomorphism,
123,
see
also
monic
morphism,
10,
see
also
map
n-category,
38
natural
isomorphism,
see
isomorphism,
natural
natural
numbers,
15,
71,
158,
see
also
arithmetic
natural
transformation,
28
composition
of,
30,
36–38
identity,
30
naturally,
32
object,
10
initial,
48,
127
as
adjoint,
49
as
limit
of
identity,
171,
173
uniqueness
of,
48
injective,
140
need
not
resemble
set,
13
probing
of,
81
projective,
140
-set
of
category,
78,
85
terminal,
48,
112,
see
also
object,
initial
open
subset,
89
order-preserving,
22,
26
ordered
set,
15,
31
adjunction
between,
54,
56,
160–162
complete
small
category
is,
162,
168
vs.
preordered
set,
16,
167
product
in,
110–111
sum
in,
128
totally,
39
partial
function,
64
partially
ordered
set,
15,
see
also
ordered
set
permutation,
39
pointwise,
23,
148,
165
polynomial,
21,
see
also
ring,
polynomial
poset,
15,
see
also
ordered
set
power,
112
182
series,
153
set,
69,
89,
110,
128
predicate,
57
preimage,
see
inverse
image
preorder,
15,
see
also
ordered
set
preservation,
see
limit,
preservation
of
presheaf,
24,
50
category
of
presheaves
is
cartesian
closed,
166
limits
in,
152
monics
and
epics
in,
156
slice
of,
157
is
topos,
169
as
colimit
of
representables,
153–156
element
of,
99
prime
numbers,
153
product,
108,
111
associativity
of,
151
binary,
111
commutativity
of,
151
empty,
111
functoriality
of,
139
informal
usage,
109
map
into,
145,
153
as
pullback,
115
uniqueness
of,
109
projection,
108,
118
projective
object,
140
pullback,
114
vs.
equalizer,
124
of
monic,
125,
135
pasting
of
pullbacks,
124
square,
115
pushout,
130,
see
also
pullback
quantifiers
as
adjoints,
57
quotient,
132,
134
of
set,
70,
129
reflection
(adjunction),
57
reflection
of
limits,
136
reflective,
46
relation,
128,
see
also
equivalence
relation
representable,
see
functor,
representable
representation
of
functor,
84,
89
as
universal
element,
99–102
of
group
or
monoid
linear,
22,
50,
157
regular,
85,
99
ring,
2
category
of
rings,
11
epics
in,
134
Index
is
not
essentially
small,
77
isomorphisms
in,
12
limits
in,
121,
137–140
is
locally
small,
76
monics
in,
123
free,
87
of
functions,
22,
89
polynomial,
8,
19,
87
SAFT
(special
adjoint
functor
theorem),
163
sameness,
33–34
scheme,
21
section,
71
sequence,
71,
92
set
axiomatization
of
sets,
79–82
category
of
sets,
11,
67
coequalizers
in,
129
colimits
in,
131
epics
in,
133
equalizers
in,
70,
113
is
not
essentially
small,
76
isomorphisms
in,
12
limits
in,
120
is
locally
small,
75
monics
in,
123
products
in,
47,
68,
107,
109
pushouts
in,
130
sums
in,
68,
127
as
topos,
82,
167
conflicting
meaning
in
ZFC,
80
definition
of,
71–73
empty,
67,
72
finite,
35,
76
of
functions,
47,
69,
164
history,
78–82
intuitive
description
of,
66
one-element,
1,
67,
112
open,
89
quotient
of,
70,
129
size
of,
74–75
structurelessness
of,
66
two-element,
69,
89,
167
-valued
functor,
84
weakly
initial,
162,
171–173
shape
of
diagram,
118
of
generalized
element,
92
sheaf,
24,
167
Sierpiński
space,
93
simultaneous
equations,
21,
113,
122
slice
category,
59
Index
of
presheaf
category,
157
small,
75,
118,
119
special
adjoint
functor
theorem,
163
sphere,
132–133
Stone–Čech
compactification,
164
string
diagram,
55
subcategory
full,
25,
103
reflective,
46
subobject,
125
classifier,
167,
168
subset,
69,
125
sum,
127,
see
also
product
empty,
127
map
out
of,
147
as
pushout,
131
supremum,
128
surface,
132–133
surjection,
133
tensor
product,
5–6,
86,
105,
165
terminal,
see
object,
terminal
thought
experiment,
120,
165,
169
topological
group,
36
topological
space,
6,
55,
see
also
homotopy
and
group,
fundamental
category
of
topological
spaces,
12
colimits
in,
137
epics
in,
134
equalizers
in,
113
is
not
essentially
small,
77
isomorphisms
in,
12
limits
in,
121,
137
is
locally
small,
76
products
in,
109
compact
Hausdorff,
122,
164
discrete,
4,
47,
87
functions
on,
22,
24,
89
Hausdorff,
134
indiscrete,
7,
47
open
subset
of,
89
subspace
of,
113
as
topos,
167
two-point,
89
topos,
82,
167–169
total
order,
39
transpose,
42
triangle
identities,
52,
56
2-category,
38
type,
79–81
underlying,
18
union,
68,
128
183
as
pushout,
130
uniqueness,
1,
3,
31,
105
of
constructions,
10,
17,
28,
42,
94
unit
and
counit,
51
adjunction
in
terms
of,
52,
53
injectivity
of
unit,
63
unit
as
initial
object,
60–63,
100
universal
element,
100
enveloping
algebra,
43
property,
1–7
determines
object
uniquely,
2,
5
upper
bound,
128
van
Kampen’s
theorem,
7,
131
variety,
36
vector
space,
3,
4,
40,
see
also
bilinear
map
category
of
vector
spaces,
12
is
not
cartesian
closed,
165
colimits
in,
137
epics
in,
134
equalizers
in,
114
is
not
essentially
small,
76
limits
in,
121,
123,
137–140
is
locally
small,
76
monics
in,
123
products
in,
110
sums
in,
127
direct
sum
of
vector
spaces,
110,
128
dual,
24,
32
free,
20,
43,
87
unit
of,
51,
58,
100
functions
on,
24
of
linear
maps,
23
vertex,
118,
126
weakly
initial,
162,
171–173
well-powered,
168
word,
19
Yoneda
embedding,
90,
102–103
does
not
preserve
colimits,
153
preserves
exponentials,
168
preserves
limits,
152
Yoneda
lemma,
94
for
monoids,
99
Z
(integers)
as
group,
39,
83,
101,
103
as
ring,
2,
48
ZFC
(Zermelo–Fraenkel
with
choice),
79–82